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\begin{document}
\bibliographystyle{amsplain}
\relax
\title{ On the local geometry of a bihamiltonian structure }
\author{ Israel~M.~Gelfand }
\author{ Ilya Zakharevich }
\address{ Dept. of Mathematics, Rutgers University, Hill Center, New
Brunswick, NJ, 08903}
\email{igelfand\atSign{}math.rutgers.edu}
\address{ Dept. of Mathematics, MIT, Cambridge, MA, 02139 }
\email{ilya\atSign{}math.mit.edu}
\date{ August 1992\quad Printed: \today }
\setcounter{section}{-1}
\maketitle
\begin{abstract}
We give several examples of bihamiltonian manifolds and show
that under very mild assumptions a bihamiltonian structure in ``general
position'' is locally of one of these types. This shows, in particular,
that a bihamiltonian manifold in general position is always a moduli
space of some kind. In the even-dimensional case it is a Hilbert scheme
of a surface, in the odd-dimensional case it is a subcotangent bundle of a
moduli space of rational curves on a surface.
\end{abstract}
\tableofcontents
\section{Introduction }
Here we want to discuss a little bit nonstandard (but becoming now
more customary) way to ask questions about an integrable system. The
usual way is to consider solutions of a, say, system
\begin{equation}
u^{\text{.}} =u'''+6uu'+u^{3},\qquad u^{\text{.}} =\frac{\partial}{\partial t}u,\quad u'=\frac{\partial}{\partial x}u,
\label{equ1}\end{equation}\myLabel{equ1,}\relax
and to study the behavior of these solutions as functions of $ x $ and $ t $.
However, it is possible to consider a manifold $ M $ of functions $ u=u\left(x\right) $
instead. In this case the equation~\eqref{equ1} determines a time evolution of a
point $ u $ of the manifold, i.e., a vector field $ V $ on this manifold.
In this consideration we lose the information of $ x $-dependence of
a solution: the notion of manifold includes the freedom to work in any
coordinate system, and values of the function $ u=u\left(x\right) $ at particular points $ x_{i} $
are in this approach just {\em some\/} coordinate functions on our manifold $ M $. So
the differential-geometric freedom of considering systems up to a
diffeomorphism results in a big restriction on the questions we can ask
about the system. However, the usual duality results in the fact that a
restriction on possible questions means the possibility to give more
precise answers on remaining questions.
Here we want to investigate the bihamiltonian geometry of the
systems in question. Let us begin with the above example. It is known that
it is possible to introduce a
couple of {\em Poisson structures\/}\footnote{Let us remind that a {\em Poisson structure\/} is a Lie algebra structure
\begin{equation}
f,g\mapsto\left\{f,g\right\}
\notag\end{equation}
on the
set of functions on the manifold satisfying the Leibniz condition:
\begin{equation}
\left\{f,gh\right\}=g\left\{f,h\right\}+\left\{f,g\right\}h.
\notag\end{equation}
} (or {\em Poisson brackets\/}) $ \left\{,\right\}_{1} $, $ \left\{,\right\}_{2} $ on the
manifold $ M $ in such a way
that the vector field $ V $ is Hamiltonian with respect to any linear
combination of the Poisson structures. That means that
the
linear combination $ \left\{,\right\}_{\lambda_{1},\lambda_{2}}= \lambda_{1}\left\{,\right\}_{1}+\lambda_{2}\left\{,\right\}_{2} $ of the Poisson brackets
\begin{equation}
\left\{f,g\right\}_{\lambda_{1},\lambda_{2}}= \lambda_{1}\left\{f,g\right\}_{1}+\lambda_{2}\left\{f,g\right\}_{2}
\notag\end{equation}
is again a Poisson bracket, and
there exist a
family of {\em Hamiltonians\/} $ H_{\lambda_{1},\lambda_{2}} $ such that for any function $ f $ on $ M $ and for
almost any pair $ \left(\lambda_{1},\lambda_{2}\right) $ (in the case of a periodical function $ u $ we can take
$ \left(\lambda_{1},\lambda_{2}\right)\not=0 $)
\begin{equation}
Vf= \lambda_{1}\left\{H_{\lambda_{1},\lambda_{2}},f\right\}_{1}+\lambda_{2}\left\{H_{\lambda_{1},\lambda_{2}},f\right\}_{2}.
\label{equ0.2}\end{equation}\myLabel{equ0.2,}\relax
(The latter formula is a particular case of the notions of the
{\em Hamiltonian flow\/} or
Hamiltonian vector
field $ V_{\varphi} $ corresponding to a function $ \varphi $, which is given by the rule
\begin{equation}
V_{\varphi}f = \left\{\varphi,f\right\},
\notag\end{equation}
hence this two conditions together mean that our vector
field is Hamiltonian with respect to any linear
combination of Poisson structures.)
Now it is easy to see that these properties of the dynamical system
in question can be cut in two: we have a condition on the Poisson
structures which has no reference to the vector field $ V $, and an
additional condition on $ V $. It is known that the former condition allows
us to find the vector field $ V $ basing on the Poisson structures in
an
almost
unique way. Hence we have two different geometrical problems: to find
the local forms (up to a diffeomorphism) of pairs of Poisson brackets
such that any linear combination is again Poisson, and
to find all the vector fields that satisfy the condition~\eqref{equ0.2}. Let us
call a pair of Poisson
structures such that any linear combination is again Poisson as {\em compatible
Poisson structures\/} or a {\em bihamiltonian structure}. In the same way we can
define $ k $-{\em hamiltonian structures}.
Although in what follows we do not need the explicit formulae for
the Poisson structures in specific coordinate frames, we give here
these formulae in
the case of the system~\eqref{equ1}. The Poisson structures are defined as
\begin{equation}
\begin{aligned}
\left\{f,g\right\}_{1} & =\int \frac{\partial f}{\partial u\left(x\right)}\left(\frac{d}{dx}\frac{\partial g}{\partial u\left(x\right)}\right)dx,
\\
\left\{f,g\right\}_{2} & =\int \frac{\partial f}{\partial u\left(x\right)}\left(\left(\left(\frac{d}{dx}\right)^{3}+4u\left(x\right)\frac{\partial}{\partial x}+2u'\left(x\right)\right)\frac{\partial g}{\partial u\left(x\right)}\right)\text{dx.}
\end{aligned}
\label{equ0.3}\end{equation}\myLabel{equ0.3,}\relax
Here $ f $ and $ g $ are functionals depending on $ u=u\left(x\right) $, $ \frac{\partial f}{\partial u\left(x_{0}\right)} $ is a
density of the partial derivative of $ f $ with respect to the coordinate
$ u\mapsto u\left(x_{0}\right) $:
\begin{equation}
f\left(u+\delta u\right)-f\left(u\right)\approx\int\delta u\left(x\right)\frac{\partial f}{\partial u\left(x\right)}dx.
\notag\end{equation}
More precisely, the left-hand side of this formula is a linear functional
in $ \delta u $, hence it
can be expressed as an integral of $ \delta u $ with some density, which we call
by definition the partial derivative $ \frac{\partial f}{\partial u\left(x\right)} $. In all these formulae
we take the integrals over the set of
definition of the function $ u $: if we consider periodical functions---along the
period, in the case of rapidly decreasing functions---along the real line.
In what follows we often interpret a Poisson
structure as a bivector field, i.e., a section $ \eta $ of $ \Lambda^{2}TM. $ Indeed, the
Leibniz condition implies that $ \left\{f,g\right\}|_{x} $ depends only on $ df|_{x} $ and $ dg|_{x} $,
hence it can be written as
\begin{equation}
\left< \eta|_{x},df\wedge dg|_{x} \right>,
\notag\end{equation}
for some $ \eta\in\Gamma\left(\Lambda^{2}TM\right) $. For example, $ \frac{\partial}{\partial x}\wedge\frac{\partial}{\partial y} $ corresponds to a Poisson
structure
\begin{equation}
\left(f,g\right)\mapsto\frac{\partial f}{\partial x}\cdot\frac{\partial g}{\partial y} -\frac{\partial g}{\partial x}\cdot\frac{\partial f}{\partial y}.
\notag\end{equation}
Now the geometrical formulation of the problem in question is:
\begin{problem} Find the normal forms for a manifold with two compatible Poisson
structures. \end{problem}
Even in the particular case of the system~\eqref{equ1} this problem seems
to be
intractable. We mean that it seems to be very difficult to determine when
the local bihamiltonian structures in neighborhoods of two given points
$ u_{1} $, $ u_{2} $ of the manifold $ M $ are diffeomorphic. We discussed some partial
results and the related analytical complications in the paper
\cite{GelZakh89Spe}.
However, the finite-dimensional variant is much simpler. In this
case (as well as in the infinite-dimensional case, in fact, see
\cite{GelZakh89Spe,GelZakhMSRI}) we should separate two cases: the case of an
even-dimensional manifold, and the case of an odd-dimensional manifold.
The reason for this is very simple: a generic skewsymmetric bilinear form
on an even-dimensional space is non-degenerate, and visa versa in the
odd-dimensional case. The generic approach to skew forms in an odd-dimensional
vector space is to take
a quotient by the kernel of the form. However, in the case of {\em a pair\/} of
forms it is not generally possible, as shows the classification theorem
from the section~\ref{h3}: the generic case of a pair of forms in an
odd-dimensional space is undecomposable, hence it should be considered
separately.
We separate here a simple and robust geometrical
classification of bihamiltonian manifolds, that occupies the sections
~\ref{h1} and~\ref{h2}, and more subtle or more technical topics, which occupy the
appendices in the sections~\ref{h3}--\ref{h6}. In turn, we separate the discussions
of the even-dimensional and the odd-dimensional cases into sections
~\ref{h1} and~\ref{h2} respectively.
In the even-dimensional case we proceed as following: in the section
~\ref{s1.1} we consider the trivial case of a $ 2 $-dimensional manifold. In the
section~\ref{s1.2} we state a particular simple case of the
unfortunately-not-so-well-known theorem on linear algebra (the complete
formulation of
this theorem appears in the section~\ref{h3}). In the section~\ref{s1.3} we, using the
constructions from the previous two sections, decompose a simplest
particular case of bihamiltonian manifold into a direct product of
(canonically defined) $ 2 $-dimensional manifolds. To do this we introduce a
notion of a {\em weak leaf\/} (that is a symplectic leaf of some linear combination
of two hamiltonian structures) and associate to a point on $ 2n $-dimensional
bihamiltonian manifold the $ n $-tuple of weak leaves passing through this
point. In the particular case we consider in this section
the set of weak leaves breaks into $ n $ (2-dimensional) connected parts, and
this $ n $-tuple contains exactly one leaf from
any parts, so we identify the manifold with the product of this $ n $
parts. In other words, in this case we can order this (generally speaking)
unordered $ n $-tuple.
In the section~\ref{s1.4} we
state a program of generalization of this construction to a less
restricted case, where we cannot separate the set of weak leaves into $ n $
parts. Therefore instead of a direct product we need to consider a
symmetric product, i.e., a quotient of a power of the set of weak leaves
by the action
of the symmetric group. In the section~\ref{s1.5} we give the definition of a
{\em regular\/} point on a bihamiltonian manifold, that is the main restriction
on the the bihamiltonian structures we can classify. Under this
restriction the set of weak leaves is a smooth $ 2 $-dimensional manifold
with a canonically defined bihamiltonian structure. In the section~\ref{s7}
we introduce the notion of a {\em Hilbert scheme}, that is a particular substitute
for a (singular)
quotient by a symmetric group. We also define a bihamiltonian structure on
the Hilbert scheme of a bihamiltonian surface. In fact up to this point
we are concerned only with definitions of the objects we need, not with
proving any meaningful theorem.
At last, in the concluding section~\ref{s1.7} of the part~\ref{h1} we combine
the introduced earlier definitions and show that the natural mapping
from the bihamiltonian manifold to the symmetrical power of the set of
weak leaves can be lifted to the mapping from this manifold to the Hilbert
scheme of the set
of weak leaves. We show that this mapping is compatible with
bihamiltonian
structures, and is a diffeomorphism under very mild conditions. We
formulate here a general theorem on the necessary conditions and a
particular case when these conditions can be geometrized: the case of a
regular point. In the latter part we use some flatness results, however we
postpone the proof of these results until the section~\ref{h4}.
At this point we drop the discussion of the even-dimensional case
and switch to the odd-dimensional case. This topic occupies the section
~\ref{h2}. However, we continue this discussion in the sections~\ref{h5} and~\ref{h6},
where we discuss more subtle properties of even-dimensional bihamiltonian
manifolds.
As in the section~\ref{h1}, we begin the discussion of the
odd-dimensional case with the corresponding part of the
theorem on the linear algebra from the section~\ref{h3}. This occupies the
section~\ref{s2.1}. In the section~\ref{s2.2} we study the geometry of the set of
weak leaves. As in the even-dimensional case, this set is
$ 2 $-dimensional.
However, now through a given point on the bihamiltonian manifold there
passes a whole $ 1 $-parametric family of weak leaves. Therefore we can
associate with a point of the bihamiltonian manifold a {\em rational curve\/} on
the surface of weak leaves. Moreover, we show that a whole family of points
corresponds to the same curve on this surface.
This motivates the definition of the Veronese web in the section
~\ref{s2.3}. It is the result of gluing together the points on the
bihamiltonian manifold that correspond to the same curve on the space of
weak leaves. After taking this quotient we lose the information on the
bihamiltonian structure, however we preserve the information on the
relative position of weak leaves.\footnote{It is the place to note that glued together points on the bihamiltonian
manifold have the same {\em action\/} variables, but different {\em angle\/} variables.
Here we use the usual in the theory of integrable systems distinction
between two sets of coordinate functions: the former can be obtained in a
simple geometrical way (compare with {\em local Hamiltonians\/} and the
{\em Magri\/}---{\em Lenard scheme\/}), the later
demand some additional work. In the case of bihamiltonian manifolds
the Lenard scheme gives a natural way to construct
the action variables. We should note that in the
odd-dimensional case the number
of angle variables is one less than the number of action variables.
\endgraf
We see here that the Veronese web reflects the geometry of the action
variables. The implications of the classification theorem for the
Veronese webs in the theory of integrable systems should be found
elsewhere. In particular, it is possible to show the existence of a
global coordinate system for the open Toda lattice such that two Poisson
structures become translation-invariant. That implies, in particular, the
existence
of a $ \frac{n\left(n-1\right)}{2} $-hamiltonian structure on the Toda lattice with $ n $ points.}
Later, in the section~\ref{s2.4}, we show
that we can reconstruct the initial bihamiltonian manifold (up to a
diffeomorphism) basing on this structure on the corresponding Veronese
web. This reconstruction uses the notion of the {\em subcotangent\/} bundle to a
Veronese web, which is turn is constructed using basic methods in the
theory of $ G $-structures. In the same way as the cotangent bundle carries a
natural Poisson structure, the subcotangent bundle carries a
natural bihamiltonian
structure. However, the construction of this structure is so complicated,
that we omit it here. The reason for these complications is the absence of
the corresponding notion of a symplectic manifold in this case, so anyone
who tried once to construct a Poisson structure on the cotangent bundle
without any reference to a symplectic structure would appreciate
these difficulties.
There is yet another part of this theorem that we leave
without a proof: the coincidence of the bihamiltonian structure on the
subcotangent bundle to a Veronese web with the bihamiltonian structure we
got this web from. However, this proof uses so unusual cohomological
construction that we do allow ourselves to present it there with several
loose ends. What remained is the definition of the {\em double cohomology\/} in the
section~\ref{s2.5}.
In the following section~\ref{s2.6} we harvest the fruits of living in
the category of analytic manifolds: we exploit a notion of the {\em twistor
transform}. To explain this notion let us consider a family of
submanifolds $ B_{\gamma}\subset B $ parameterized by a manifold $ \Gamma\ni\gamma $. We can consider the
universal family $ A=\left\{\left(b,\gamma\right) \mid b\in B_{\gamma}\right\}\subset B\times\Gamma $ and two projections
\begin{equation}
\begin{matrix}
& & A
\\
& \swarrow & & \searrow
\\
B & & & & \Gamma.
\\
\end{matrix}
\notag\end{equation}
We can see that the notion of a family of submanifolds is a particular
case of a notion of a double bundle. However, in the latter notation $ B $
and $ \Gamma $ appear in the same form, so we can interchange them and instead of
a family of submanifolds in $ B $ parameterized by $ \Gamma $ we can consider a family
of submanifolds in $ \Gamma $ parameterized by $ B $. To a point $ b\in B $ we associate a
submanifold $ \Gamma_{b}\subset\Gamma $ consisting of passing through $ b $ members of the former
family.
This is still not a definition of a twistor transform, however we
call a family $ \Gamma_{b} $ a twistor transform of a family $ B_{\gamma} $ if the geometry of
the former family is somewhat simpler than the geometry of the latter.
This is the case for Veronese webs. By definition a Veronese web is a
family of hypersurfaces, so we can consider the twistor transform, that
is a family of rational curves on a surface. Moreover, we show that this
family is in fact a {\em complete family\/}: it contains any deformation of any
curve in this family. That means that we need not specify the family: it
is determined by the geometry of the surface itself. Therefore {\em to
determine a Veronese web it is sufficient to provide a surface with a
rational curve that allows a deformation}.
In the section~\ref{s2.7} we discuss shortly the consequences of the
twistor
transform in the theory of bihamiltonian manifolds and Veronese webs,
like the amount of non-diffeomorphic bihamiltonian systems and the
fusion operations over Veronese webs. At last, in the sections~\ref{s2.8} and
~\ref{s2.9} we consider the simplest non-trivial cases: $ 2 $-dimensional Veronese
webs and $ 3 $-dimensional bihamiltonian manifolds.
At this point we leave the discussion of basic geometrical questions
and delve into details and technicalities. As we have already mentioned,
in the section~\ref{h3} we state the general theorem on classification of
pairs of linear mappings or bilinear forms. Even if technical, this
theorem
concerns the utterly basic notions that by strange reasons are dropped in
the basic mathematical education. Though we state the necessary
particular cases of this theorem in the places we use them, we strongly
recommend the reader {\em to begin\/} with reading this appendix. Furthermore, we
want to note that this theorem is a keystone in our approach to the
concerned here problems, and a generalization of this theorem to the
infinite-dimensional case of the forms~\eqref{equ0.3} associated with the KdV
equation was the
challenge that eventually piloted us to the questions concerned here.
In the section~\ref{h4} we provide the missed link in the proof of the
existence of the mapping to the Hilbert scheme.
In the section~\ref{h5} we consider the global geometry of an
even-dimensional bihamiltonian manifold. We construct here examples of
manifolds such that the topology itself defines a polyhamiltonian
structure on these manifolds. Moreover, as we will see in the section
~\ref{s6.4}, these manifolds give as example of compact manifolds such that
the weak classification theorem is applicable to them in any point.
In the final section~\ref{h6} we exploit the classification theorem to
investigate the local geometry of a bihamiltonian manifold. We begin the
section~\ref{s6.1} with an example of the case when the weak classification
theorem is applicable, but the strong one is not. It is an example of a
non-regular point on the Hilbert scheme $ S^{2}{\mathbb A}^{2} $, and we make the preparations to
find the set of regular point on a generic Hilbert scheme. We give a
description of the tangent space to a Hilbert scheme, but we fail to
describe a bivector that corresponds to a Poisson structure. However, we
can give a formula for such a bivector if we know the
bivector that
corresponds to some other Poisson structure. This information is
sufficient to describe the {\em recursion operator\/} of the bihamiltonian
structure. We do it in the section~\ref{s6.2}.
In the section~\ref{s6.3} we use these results to give the description of
the subset of regular points on a Hilbert scheme. Here we also describe
generalized weak leaves, that allows us to show in the section~\ref{s6.4} that
the weak classification theorem is applicable in any point of a Hilbert
scheme. We also give here an example of a weak leaf with a singular
closure and study a tangent cone to this closure.
In the section~\ref{s6.5} we introduce a natural identification
of a
neighborhood of a regular point on a Hilbert scheme with an open subset
in the cotangent bundle to polynomials of one variable. This determines
another Poisson structure on this bundle, and it has polynomial
coefficients. This identification shows also that the conditions of the
Magri Classification theorem for bihamiltonian structures \cite{Mag} are
satisfied in this case, that
shows essentially that the strong classification theorem and the Magri
classification theorem are in fact applicable in absolutely the
same cases (if we modify the Magri theorem a little). However, we want to note
that this equivalence is obtained by
providing an isomorphism of {\em models\/} of bihamiltonian manifolds, not by
comparing the conditions of these theorems---they remain still absolutely
unrelated.
While the Magri coordinate system gives a model in which one Poisson
structure is constant and another polynomial, in the section~\ref{s6.6} we
introduce another coordinate system on the Hilbert scheme. In this
coordinate system the first
Poisson structure remains constant, but the second structure becomes
{\em linear}. That means that the geometry in a neighborhood of a regular point
is connected with a particular (finite-dimensional) Lie algebra with two
cocycles. This algebra is described in the final section~\ref{s6.7}. Let us
note that the knowledge that a bihamiltonian structure can be written in
such a form could allow one to find the corresponding classification
theorems by solving some standard problems of linear algebra, like: {\em find
all the possible structure constants for a\/} $ Li $ {\em algebra structure for
which two given bilinear forms are cocycles}. We solve the simplest
non-trivial particular case of this problem in the section~\ref{s6.8}. The
solution provides us with a rich set of examples of $ 4 $-dimensional
bihamiltonian systems. In two of these examples the {\em set of weak leaves\/}
$ M^{\left(2\right)} $ {\em is non-smooth!\/} This example shows that we cannot drop one condition
on the weak classification theorem. However, in the same section we show
that in this particular case the theorem remains true if we consider a
{\em normalization\/} of the Hilbert scheme instead of the Hilbert scheme itself.
That gives a hope to generalize the theorem on this direction.
We want to emphasize here the unexpected similarity between the
even-dimensional and odd-dimensional cases: in both cases we can
reconstruct the initial bihamiltonian manifold basing on the surface of
weak leaves. And in both cases the reconstruction involves taking the
moduli spaces of submanifolds on this surface, though the dimensions of
these submanifold are different.
We are indebted to a lot of people for fruitful discussions and
inestimable help, among them A.~Givental, A.~Goncharov,
M.~Kontsevich, F.~Magri,
A.~Radul, N.~Wallach and A.~Weinstein. We should express special thanks to
Henry McKean who provided
us with his variant of the Magri Classification theorem \cite{McKeanPC}, which
inspired the discussions of the even-dimensional case here, and to Vera
Serganova, whose patient remarks allowed this paper to acquire its
current form. The idea to use the twistor transform for the local
classification of the Veronese webs is due to A.~Goncharov \cite{GonPC}.
Anywhere in this paper (if not stated otherwise) we consider
analytic manifolds. However, a lot of results can be easily translated to
the $ C^{\infty} $-case.
\section{The even-dimensional case }\label{h1}\myLabel{h1}\relax
\subsection{A $ 2 $-dimensional example }\label{s1.1}\myLabel{s1.1}\relax
\begin{example} Let us try to classify
$ 2 $-dimensional bihamiltonian systems in
general position. In dimension two {\em any\/} bivector field corresponds to a
Poisson structure, so we should simply classify pairs of bivector fields.
We can suppose that at the given point the bivector
corresponding to the first Poisson structure is non-degenerate. That
means that in a neighborhood of this point this Poisson structure is ``an
inverse'' of a symplectic structure. We can chose a local coordinate
system $ \left(x_{1},x_{2}\right) $ such that this symplectic structure can be written as
$ dx_{1}\wedge dx_{2} $, hence the Poisson structure can be written as $ \frac{\partial}{\partial x_{1}}\wedge\frac{\partial}{\partial x_{2}} $.
Now we can consider a ratio of our bivector fields, which is a
function on $ M $. If the given point is not a critical point of this
function, then we can chose it as the first coordinate $ x_{1} $ and can still
find another function $ x_{2} $ such that the first Poisson structure is
$ \frac{\partial}{\partial x_{1}}\wedge\frac{\partial}{\partial x_{2}} $. Hence the second Poisson structure is $ x_{1}\frac{\partial}{\partial x_{1}}\wedge\frac{\partial}{\partial x_{2}} $.
We see that in $ 2 $-dimensional case there is essentially {\em one\/}
bihamiltonian manifold, and any manifold in general position is locally
isomorphic to {\em some\/} part of this manifold (that is a small difference
with d'Harboux theorem, where any symplectic manifold is isomorphic to
{\em any\/} part of the fixed manifold).
In the case of not general position the classification is reduced to
(well-known) problem of local classification of a function on a symplectic
manifold. \end{example}
This example seems to be trivial, however it implies a powerful
construction in a multidimensional case: if $ M_{1} $ and $ M_{2} $ are two
bihamiltonian manifolds, then $ M_{1}\times M_{2} $ is also a bihamiltonian manifold. In
this way using the example above we can construct a bihamiltonian
manifold of any even dimension. Now we can ask if the universality
property of $ 2 $-dimensional example is still true in this case. The answer
is the following particular case of the Turiel theorem \cite{Tur89Cla}:
\begin{theorem} \label{th1}\myLabel{th1}\relax Consider a point $ m $ on an $ 2n $-dimensional bihamiltonian
manifold $ M $. If a couple of bivectors $ \eta_{1}|_{m},\eta_{2}|_{m}\in\Lambda^{2}T_{m}M $ is in general
position, then in a neighborhood of $ m $ it is possible to choose
a coordinate system
in such a way that the bivector fields are
\begin{equation}
\eta_{1}=\sum_{k=1}^{n}f_{k}\left(x_{2k-1},x_{2k}\right)\frac{\partial}{\partial x_{2k-1}}\wedge\frac{\partial}{\partial x_{2k}},\quad \eta_{2}=\sum_{k=1}^{n}g_{k}\left(x_{2k-1},x_{2k}\right)
\frac{\partial}{\partial x_{2k-1}}\wedge\frac{\partial}{\partial x_{2k}}.
\notag\end{equation}
Therefore our bihamiltonian system is represented as a product of
$ 2 $-dimensional ones.
It is easy to see that if not only the values, but also the
derivatives of the bivector {\em fields\/}
$ \eta_{1} $, $ \eta_{2} $ at the point $ m $ are in general position,
we can set
\begin{equation}
\eta_{1}=\sum_{k=1}^{n}\frac{\partial}{\partial x_{2k-1}}\wedge\frac{\partial}{\partial x_{2k}},\quad \eta_{2}=\sum_{k=1}^{n}x_{2k-1}\frac{\partial}{\partial x_{2k-1}}\wedge\frac{\partial}{\partial x_{2k}},
\notag\end{equation}
if we allow the point $ m $ to correspond to an arbitrary point $ \left(x_{i}\right) $ and not
necessary to the origin
$ \left(x_{i}\right)=0 $. \end{theorem}
Hence any generic bihamiltonian manifold is locally isomorphic to a
product of $ 2 $-dimensional manifolds from the example.
In fact the Turiel theorem is much more powerful (it handles also
some cases of not general position),
however, in the following discussion we need only this particular case.
It is also quite easy to prove this case (as well as the Turiel theorem in
whole) using the general arguments of linear algebra.
\subsection{A theorem from linear algebra }\label{s1.2}\myLabel{s1.2}\relax Indeed, the linear algebra says
that a pair of skewsymmetric
bilinear forms $ \left(\alpha,\beta\right) $
in general position
in an even-dimensional vector space $ V $ can be canonically
decomposed in a direct sum of pairs of forms in $ 2 $-dimensional spaces:
\begin{equation}
V=\bigoplus_{i}V_{i},\qquad \alpha=\bigoplus_{i}\alpha_{i},\quad \beta=\bigoplus_{i}\beta_{i},
\notag\end{equation}
where $ \left(\alpha_{i},\beta_{i}\right) $ is a pair of skewsymmetric bilinear forms in $ V_{i} $
(see, for example, \cite{GelZakh89Spe}). It also
says that the forms in these subspaces are proportional with some
coefficients $ \lambda_{i} $ (we call them {\em eigenvalues\/}). The exact formulation of this
theorem can be found below, in the appendix~\ref{h3}.
\subsection{A map to the set of weak leaves }\label{s1.3}\myLabel{s1.3}\relax Let us
apply this argument to the space $ V=T_{m}^{*}M $ and to the pair of forms $ \eta_{1}|_{m} $,
$ \eta_{2}|_{m} $ in this space. We see that the cotangent space is canonically
decomposed into a direct sum of $ 2 $-dimensional subspaces. That means that
the tangent space is also decomposed into a direct sum of $ 2 $-dimensional
subspaces. The same is evidently true at nearby points. Hence there are $ n $
canonically defined distributions of $ 2 $-dimensional subspaces in the
tangent bundle of $ M $ and $ n $ functions $ \lambda_{i} $ on $ M $.
The next step is to prove that these distributions are integrable,
i.e., are the tangent bundles to some foliations. To do this we should do
the following: given a point and a fixed $ i $-th family of subspaces we
should
find a $ 2 $-dimensional submanifold satisfying the following properties:
\begin{enumerate}
\item
it passes through the given point in the
direction of the $ 2 $-dimensional subspace of the family of subspaces;
\item
the tangent space at any point of this
submanifold is a subspace from the famimly.
\end{enumerate}
Again, to do this it is
sufficient to find a
submanifold of {\em codimension\/} 2 such that the tangent space at any point of
it is a sum of $ n-1 $ marked subspaces. Now fix a point $ m\in M $ and $ 1\leq i\leq n $. The
bivector $ \eta_{1}|_{m}-\lambda_{i}\left(m\right)\eta_{2}|_{m} $ considered as a bilinear form in the cotangent
space $ T_{m}^{*}M $ has a $ 2 $-dimensional kernel by the definition of $ \lambda_{i} $. Now we
need (the first time) some information of the geometric structure of a
Poisson manifold.
\begin{definition}
\begin{enumerate}
\item
A submanifold $ L $ of a symplectic manifold $ M $ is called a
{\em Poisson submanifold\/} if the restriction of $ \left\{f,g\right\} $ on $ L $ is uniquely
determined by the restrictions of the functions $ f $ and $ g $ on $ L $. In this
case this restriction determines a Poisson structure on $ L $.
\item
A {\em symplectic leaf\/} in a Poisson manifold $ M $ is an imbedded Poisson
submanifold
$ L $ of $ M $ such that the corresponding Poisson structure on $ L $ is
{\em nondegenerate}, i.e., corresponds to some symplectic form $ \omega $ on L.\footnote{In fact, since an open subset of a symplectic leaf is again a symplectic
leaf, in what follows we restrict this definition and call as a symplectic
leaf only the {\em maximal\/} submanifolds with the specified properties.}
\end{enumerate}
\end{definition}
Well-known theorem on the local structure of a symplectic manifold
\cite{Wei83Loc} claims in particular that:
\begin{theorem} There exists a unique symplectic leaf passing through any
given point $ m $ of a Poisson manifold $ M $. The normal space to this leaf at $ m $
coincides with the kernel of the bivector $ \eta|_{m}\in\Lambda^{2}T_{m}M $ considered as a
bilinear form in the space $ T_{m}^{*}M $. \end{theorem}
Let us apply this theorem to the Poisson structure $ \eta_{1}-\lambda_{0}\eta_{2} $
on $ M $,
where $ \lambda_{0}=\lambda_{i}\left(m\right) $. The symplectic leaf
$ L_{m,i} $
passing through $ m $ has a desired
tangent space at $ m $, moreover, the normal space to it at any point $ m' $ of
this leaf is the kernel of $ \eta_{1}-\lambda_{0}\eta_{2} $. That means that $ \lambda_{i}|_{L_{m,i}}=\lambda_{0}=\lambda_{i}\left(m\right) $. Now we
can construct the desired $ 2 $-dimensional submanifold
$ \widetilde L_{m,i} $
as an intersection of $ n-1 $ submanifolds $ L_{m,j} $ for $ j\not=i $. We can also use the
constructed foliation of codimension 2 to define a local projection $ \pi_{i} $ of $ M $
to a local base $ M_{i} $ of this foliation.
\begin{definition} Let us call any symplectic leaf (of non-vanishing
codimension) of any linear combination $ \eta_{1}-\lambda\eta_{2} $ a {\em weak leaf\/} of the
bihamiltonian structure. \end{definition}
It is easy to see that in our case the set of parameters of weak
leaves $ M^{\left(2\right)} $ is $ 2 $-dimensional and is (locally on $ M $) a disjoint union of $ n $
submanifolds $ M_{i} $ corresponding to (different) eigenvalues of the pair
$ \left(\eta_{1},\eta_{2}\right) $ at the point $ m $.
\begin{remark} In this case we can {\em define\/} $ M^{\left(2\right)} $ as the union of local bases for
foliations $ \left\{L_{i}\right\} $. However, since this space plays a crucial r\^ole in what
follows we want to emphasize the fact that in general case this
(topological) space isn't even Hausdorff. If we move a point $ m $ on a
manifold the pair of forms in $ T_{m}^{*}M $ changes and a pair of eigenvalues can
collide. If they do it in a ``civilized manner'', this results in a Jordan
block of the corresponding matrix (see appendix) and the dimension of the
kernel does not change (this is exactly the case we are interested in
below). However, a collision of eigenvalues can also result in an
eigenspace of greater dimension, i.e., in a weak leaf of different
dimension.
However, in the cases we study below {\em the space of weak leaves\/} $ M^{\left(2\right)} $
is a smooth manifold, so we have no such complications. \end{remark}
Combining the above $ n $ projections we get a local identification of $ M $ and
a product of $ n $ 2-dimensional manifolds $ M_{i} $. What is remaining to prove is
that the bivector fields are products of some bivector fields on the
factors. It is sufficient to consider {\em one\/} bivector field, say, $ \eta=\eta_{1} $.
Moreover, we can suppose that the Poisson structure $ \eta $ is non-degenerate,
since we can consider two non-degenerate linear combinations of $ \eta_{1} $, $ \eta_{2} $
instead of considering $ \eta_{1} $, $ \eta_{2} $.
What follows from the choice of the projections is that for any fixed
point (say, $ m $) on $ M $ we can represent {\em values\/} of the bivector fields (i.e.,
a pair of bivectors) as two products of bivectors on the factors:
\begin{equation}
\eta|_{m}=\bigoplus_{i}\eta_{i}\left(m\right),\quad \eta_{i}\left(m\right)\in\Lambda^{2}T_{\pi_{i}\left(m\right)}M_{i},\quad T_{m}M=\bigoplus_{i}T_{\pi_{i}\left(m\right)}M_{i}.
\notag\end{equation}
If we denote the coordinates on $ M_{i} $ by $ x_{i} $, $ y_{i} $, we can express this fact by
the formula
\begin{equation}
\eta\left(x_{1},y_{1},\dots ,x_{n},y_{n}\right)=\sum_{i}\widetilde\eta_{i}\left(x,y\right) \frac{\partial}{\partial x_{i}}\wedge\frac{\partial}{\partial y_{i}}.
\label{equ3.3}\end{equation}\myLabel{equ3.3,}\relax
What we should prove is that the $ \widetilde\eta_{i}\left(x,y\right) $ depends only on $ x_{i} $, $ y_{i} $, i.e.,
that for any two point $ m' $, $ m'' $ on $ L_{m,i}=\pi_{i}^{-1}\left(\pi_{i}\left(m\right)\right) $ the $ i $-components of $ \eta $
are the same. That means that for any two functions $ \varphi_{1} $, $ \varphi_{2} $ on $ M $ that
depend only on $ x_{i} $, $ y_{i} $ the Poisson bracket also depends only on $ x_{i} $, $ y_{i} $.
\begin{remark} Let us note that this property is specific to the symplectic
geometry. In, say, the Riemannian case the coordinate foliations
$ \left(x_{i}=\operatorname{const}\right) $ can be orthogonal with respect to a Riemannian form that
cannot be represented as a direct product. \end{remark}
To prove this property we will use the second theorem on the geometry
of a Poisson manifold.
\begin{theorem}
\begin{enumerate}
\item
The Hamiltonian vector field $ V_{f} $ corresponding to a function $ f $
on a Poisson manifold $ M $ preserves the Poisson structure.
\item
The {\em fundamental relation\/} links the operations of commutation of
vector fields and the Poisson bracket on functions:
\begin{equation}
V_{\left\{f,g\right\}}=\left[V_{f},V_{g}\right].
\notag\end{equation}
\end{enumerate}
\end{theorem}
Now to prove this fact it is sufficient to construct sufficiently
many vector fields on $ M $ that preserve the Poisson structure and
are tangent to fibers of the projection $ \pi_{i} $. These fibers are spanned by the
Hamiltonian
flows corresponding to functions that depend only on $ x_{j},y_{j} $, $ j\not=i $ (by the
property~\eqref{equ3.3}). However, we can find a function $ f\left(x_{j},y_{j}\right) $, $ j\not=i $, such
that the Hamiltonian
flows corresponding to the this function moves the given point in an
arbitrary direction tangent to a fiber of $ \pi_{i} $. This
finishes the proof of the
fact that either bivector
field is a product of fields on specified $ 2 $-dimensional manifolds.
\begin{remark} We can write down the last arguments of the proof with formulae
\begin{multline} \left\{x_{j},\left\{\varphi\left(x_{i},y_{i}\right),\psi\left(x_{i},y_{i}\right)\right\}\right\} =
\\
\left\{\left\{x_{j},\varphi\left(x_{i},y_{i}\right)\right\},\psi\left(x_{i},y_{i}\right)\right\} + \left\{\varphi\left(x_{i},y_{i}\right),\left\{x_{j},\psi\left(x_{i},y_{i}\right)\right\}\right\} =0,
\notag\end{multline}
if $ i\not=j $, and the same with a change of $ x_{j} $ to $ y_{j} $
(application of~\eqref{equ3.3}). This means that $ \left\{\varphi\left(x_{i},y_{i}\right),\psi\left(x_{i},y_{i}\right)\right\} $ is
preserved by Hamiltonian flows corresponding to $ x_{j} $, $ y_{j} $, and, therefore,
is constant along the fibers of the projection $ \pi_{i} $. Here the
non-degeneracy of $ \eta $ guaranties that these Hamiltonian flows span a whole
fiber.
\end{remark}
\begin{remark} Now it is easy to see that any local automorphism of a
bihamiltonian
manifold in general position is coming from $ n $ diffeomorphisms of
$ 2 $-dimensional factors $ M_{i} $. In
particular, any vector field that is Hamiltonian with respect to both
Poisson structures is a product of such fields on the factors. So it is
again sufficient to consider the $ 2 $-dimensional case, where such a field
should preserve the ratio of the Poisson structures. Hence the
Hamiltonians of this vector field are constant on the level lines of the
ratio. From the other side, it is easy to see that the Hamiltonian flow
of such a function is Hamiltonian with respect to any non-degenerate
combination of the Poisson structures in question.
Therefore {\em a vector
field that is hamiltonian with respect to both Poisson structures is
Hamiltonian with respect to any nondegenerate linear combination of them
and the
Hamiltonian\/} $ H $ {\em of this field (with respect to any of the Poisson structures) is
a sum\/}
\begin{equation}
H=H_{1}+H_{2}+\dots +H_{n}
\notag\end{equation}
{\em of functions\/} $ H_{i} $ {\em such that either\/} $ H_{i} $ {\em depends (maybe, multivalued)
on the
eigenvalue\/} $ \lambda_{i} ${\em, or\/} $ \lambda_{i} $ {\em is constant and then\/} $ H_{i} $ {\em is constant on the\/} $ i ${\em -th
family of weak leaves.\/} \end{remark}
\begin{remark} In fact we have constructed a mapping
\begin{equation}
M\to M_{1}\times M_{2}\times\dots \times M_{n}\hookrightarrow M^{\left(2\right)}\times M^{\left(2\right)}\times\dots \times M^{\left(2\right)},
\notag\end{equation}
and to do this we have fixed an order of eigenvalues $ \left\{\lambda_{i}\right\} $. If we do not
fix this order, we can still construct a mapping to a factor by the
action of the symmetrical group
\begin{equation}
M\to \underbrace{M^{\left(2\right)}\times M^{\left(2\right)}\times\dots \times M^{\left(2\right)}}_{n\text{ times}}/{\mathfrak S}_{n}.
\notag\end{equation}
In fact the above considerations show that on $ M^{\left(2\right)} $ there is a natural
bihamiltonian structure, and if we consider a cross-product
bihamiltonian structure on $ M^{\left(2\right)}\times M^{\left(2\right)}\times\dots \times M^{\left(2\right)} $, then the former
mapping is a local isomorphism of Poisson manifolds. From the other
hand, the taking of the quotient by $ {\mathfrak S}_{n} $ behaves well with
respect to the
Poisson structure on $ M^{\left(2\right)}\times M^{\left(2\right)}\times\dots \times M^{\left(2\right)} $, and the image of the latter
mapping consists of {\em smooth\/} points on $ \underbrace{M^{\left(2\right)}\times M^{\left(2\right)}\times\dots \times M^{\left(2\right)}}_{n\text{ times}}/{\mathfrak S}_{n} $,
therefore the latter mapping is also a local isomorphism of Poisson
manifolds.
\end{remark}
We can conclude now that any (local) even-dimensional bihamiltonian
manifold
in general position is (locally) isomorphic to some domain in a fixed
bihamiltonian manifold. Hence to specify a local diffeomorphism type
of a bihamiltonian manifold in a neighborhood of a given point it is
sufficient to specify a point on that manifold, i.e. a finite number of
parameters.
\subsection{A case with non-trivial monodromy }\label{s1.4}\myLabel{s1.4}\relax Let us consider now a local
bihamiltonian manifold such that the
condition of the theorem~\ref{th1} is not satisfied. If the
manifold is
in sufficiently ``general position'' we can expect that the points where
this condition {\em is\/} satisfied form a dense open subset. Let us call a point
{\em good\/} if the condition of the theorem~\ref{th1} is satisfied at this point.
Let us call the subset of such points $ U $ (evidently, this subset is open).
In this case we cannot be sure that the set $ U^{\left(2\right)} $ of weak leaves in $ U $ is a
disjoint union of $ n $ components and that though any point of $ U $ passes a
leaf from any component of $ U^{\left(2\right)} $. It is possible to say which weak leaf
passing through a point $ x' $ (from a vicinity of a given point $ x\in U $)
corresponds to a given weak leaf passing through $ x $, but the monodromy
along a closed loop in $ U $ can interchange weak leaves passing through $ x $.
However, the map
\begin{equation}
U\xrightarrow[]{\alpha} \underbrace{U^{\left(2\right)}\times U^{\left(2\right)}\times\dots \times U^{\left(2\right)}}_{n\text{ times}}/{\mathfrak S}_{n}
\notag\end{equation}
is still well-defined and is a local isomorphism of Poisson manifolds.
Magri in his article \cite{Mag} has shown that sometimes it is possible to
classify
bihamiltonian systems even in a neighborhood of a non-good point.
In the case of general position which he has studied the manifold is
again locally isomorphic to a piece of {\em one
particular\/}
bihamiltonian manifold given by explicit formulae.
Here we want to give a geometrical reason for such a phenomenon, and this
reason is that under mild conditions the mapping $ \alpha $ can be extended to a
mapping
\begin{equation}
M\xrightarrow[]{\alpha} \underbrace{M^{\left(2\right)}\times M^{\left(2\right)}\times\dots \times M^{\left(2\right)}}_{n\text{ times}}/{\mathfrak S}_{n},
\label{equ4.3}\end{equation}\myLabel{equ4.3,}\relax
and (if we modify slightly the definition of taking the
quotient
by $ {\mathfrak S}_{n} $ to obtain
a smooth quotient) this map is a local isomorphism. Since in the case of
general position $ M^{\left(2\right)} $ is a piece of {\em one particular\/} bihamiltonian plane
$ \left({\mathbb A}^{2},\frac{\partial}{\partial x}\wedge\frac{\partial}{\partial y},x\frac{\partial}{\partial x}\wedge\frac{\partial}{\partial y}\right) $, the manifold in the right-hand side of
the formula~\eqref{equ4.3} is a piece of {\em one particular\/} bihamiltonian manifold.
Here we show that conditions under which the map~\eqref{equ4.3} exists
can be made
much more mild than the Magri conditions, hence we obtain here a
(minor) {\em generalization\/} of the Magri result. These milder conditions are,
however, not very constructive, so we give also stronger
conditions of
geometrical nature. These geometrical conditions are of very different
origin from the Magri conditions, and the resulting description is of a
bihamiltonian structures is absolutely different. We will close this gap
in the concluding the paper appendix, where we show that our
geometrical conditions (practically) coincide with the Magri conditions!\footnote{Frankly speaking, we cannot give a direct proof of this fact, we just
show that
his {\em canonical\/} form satisfies our conditions, and our canonical form
(almost always) satisfies his conditions. That means that we use power of
{\em both\/} classification theorems to show that the conditions coincide!}
So the main goal of the discussion of the even-dimensional
bihamiltonian systems is not to give a generalization of the Magri
construction, but to find a geometrical framework allowing to
construct a {\em canonical\/} identification of the bihamiltonian manifold with a
basic example of such a manifold.
Now we want to list the conditions under which the mapping is
well-defined and well-behaved.\footnote{In fact there is a big confidence that a suitable {\em algebraization\/} of the
following discussion can help in weakening the conditions we specify,
however, we want here to use a {\em synthetic\/} language and work with smooth
manifolds wherever it is possible.} First of all we want to list the
obstruction for this mapping to exist.
We want to repeat it again that the variety $ M^{\left(2\right)} $ in the general
case can be
non-smooth and even have ``components'' of different dimensions. If
we live in a vicinity of a good point then the mapping
\begin{equation}
M\xrightarrow[]{\alpha} \underbrace{M^{\left(2\right)}\times M^{\left(2\right)}\times\dots \times M^{\left(2\right)}}_{n\text{ times}}/{\mathfrak S}_{n}
\notag\end{equation}
sends a point of $ M $ to a point of the quotient that corresponds to an
$ n $-tuple of different points of $ M $. In such points the smooth structure on the
quotient is well-defined. However, if we move to the {\em boundary\/} of the
good set $ U $ the eigenvalues {\em collide\/} and to a point of $ M $ can correspond
an $ n $-tuple of points of $ M^{\left(2\right)} $ where some points can appear with
some {\em multiplicity}. It is known (see the example below) that the variety
$ \underbrace{M^{\left(2\right)}\times M^{\left(2\right)}\times\dots \times M^{\left(2\right)}}_{n\text{ times}}/{\mathfrak S}_{n} $ is {\em singular\/} at such points. Therefore we
need some {\em desingularization\/} $ S^{n}M^{\left(2\right)} $ of this variety, and this
desingularization
should be sufficiently small\footnote{If we consider a non-degenerate bivector field on a manifold $ M $ and a
blow-up $ \widetilde M $ of this manifold in some submanifold $ N $, then the
corresponding bivector field on $ \widetilde M $ has a pole on the preimage of $ N $.
Therefore we cannot make any additional blow-ups on the manifold we want
to construct.} to extend the Poisson structure from smooth
points of $ \underbrace{M^{\left(2\right)}\times M^{\left(2\right)}\times\dots \times M^{\left(2\right)}}_{n\text{ times}}/{\mathfrak S}_{n} $. If this desingularization {\em is\/}
sufficiently small we can expect that the map $ \alpha $ can be raised to a map
$ M\to S^{n}M^{\left(2\right)} $, and we need this map to be a local diffeomorphism. Now {\em the
result of Magri shows that in proper cases this program can be
fulfilled\/}!
Here we just list the ``algebraic nonsense'' that allows to fulfill it
under very mild conditions.
Let us list here the assumptions we need to make the Magri result
``{\em functorial\/}'':
\begin{enumerate}
\item
We need the weak leaves to have a good parameter space $ M^{\left(2\right)} $ on the
whole $ M $;
\item
We need a good desingularization $ S^{n}M^{\left(2\right)} $ of $ \underbrace{M^{\left(2\right)}\times M^{\left(2\right)}\times\dots \times M^{\left(2\right)}}_{n\text{ times}}/{\mathfrak S}_{n} $;
\item
We need a map $ M\to S^{n}M^{\left(2\right)} $;
\item
We need a bihamiltonian structure on $ S^{n}M^{\left(2\right)} $.
\end{enumerate}
\subsection{A case of a regular point }\label{s1.5}\myLabel{s1.5}\relax If we consider a bihamiltonian manifold in
general position, then
the points where the above analysis is applicable form a dense open
subset. In fact the theorem on linear algebra from the appendix defines
{\em the eigenvalues\/} even outside of this subset. However, we can define the
eigenvalues much more
simple. Indeed, a bivector field $ \eta $ determines a mapping $ \widetilde\eta_{m}\colon T_{m}^{*}M\to T_{m}M $ for
any point $ m\in M $, and eigenvalues in question are just eigenvalues\footnote{The theorem on linear algebra shows that to any eigenvalue of a pair of
forms $ \eta_{1} $, $ \eta_{2} $ in the above sense corresponds a {\em double\/} eigenvalue of the
recursion operator.} of the
{\em recursion\/} map
\begin{equation}
\widetilde\eta_{1,m}^{-1}\widetilde\eta_{2,m}\colon T_{m}^{*}M\to T_{m}^{*}M
\notag\end{equation}
that is defined anywhere where $ \eta_{1} $ is
non-degenerate. Hence the complement to this dense subset consists of
points where the eigenvalues collide.
Let us consider the first question first. A passing through $ m\in M $ weak
leaf corresponds to a kernel of a bilinear form $ \eta_{1}-\lambda\eta_{2} $ on the space $ T_{m}^{*}M $
(since, say, the theorem on a local structure of a Poisson
manifold \cite{Wei83Loc} applied to $ \eta_{1}-\lambda\eta_{2} $ shows that there is a weak leaf with
this
kernel as a normal space). So to have a good parameter space of weak
leaves we need at least the leaves to have the same dimension, i.e., any
linear combination $ \eta_{1}-\lambda\eta_{2} $ to have at any point $ m\in M $ at most $ 2 $-dimensional
kernel (that guaranties that any weak leaf is of codimension 2).\footnote{However, below we give a definnition of a generalized weak leaf that
allows to drop this restriction.}
The theorem on the structure of a pair of skewsymmetric bilinear
forms \cite{GelZakh89Spe} (or see in the appendix on linear algebra) shows that in
this case the corresponding pair of linear mappings has only one Jordan
block for any eigenvalue. It is clear that the set of pairs satisfying
this condition is open and that the {\em stabilizer\/} of any such pair has the
same dimension as the stabilizer of a pair in general position. Therefore
it is a closest generalization of the notion of {\em a pair in general
position}.
\begin{definition} Let us call a pair of skew-symmetric bilinear forms in a
vector space $ V $ a {\em regular pair}, if the stabilizer of this pair in $ \operatorname{GL}\left(V\right) $
has minimal possible dimension.
Let us call a point $ m $ of bihamiltonian manifold $ M $ a {\em regular point}, if
the corresponding pair of bilinear forms in $ T_{m}^{*}M $ is a regular pair.
\end{definition}
So a regular pair in an even-dimensional space corresponds to a pair
of mappings
that has exactly one Jordan block for any eigenvalue and no Kroneker
blocks at all.
It is clear that the set of regular points is open, hence any weak
leaf passing through a vicinity of a regular point is of codimension 2.
If any leaf intersects with the set of good points, then space of weak
leaves is smooth in the corresponding point.\footnote{In this case the parameter space of weak leaves does not change if we
consider only the
space of good points, and the parameter space {\em is\/} smooth in the latter
case.} Since the eigenvalue $ \lambda $ is
constant on a leaf, to
satisfy this condition it is sufficient to demand that the set of
not-good points with a given eigenvalue is of codimension at least 3.
\begin{remark} Consider the two defined above Poisson brackets on an
open subset of the set of weak leaves. If any weak leaf intersects the
set of good points, then these Poisson
brackets can obviously be extended to the whole space of weak leaves. It
would be very interesting to understand if this fact is true in the
general case (including the generalization on the case of generalized
weak leaves). Compare the theorem~\ref{th1.5}. \end{remark}
\subsection{A good symmetrical power }\label{s7}\myLabel{s7}\relax So the first condition is explained. What
is the meaning of the
second condition? The problem with a definition of $ S^{n}M^{\left(2\right)} $ is that the
quotient
by an action of a group can be singular.
\begin{example} \label{ex7.1}\myLabel{ex7.1}\relax Let us consider the action of $ {\mathbb Z}_{2} $ on a plane $ \left(x,y\right) $ by
reflection $ \left(x,y\right)\mapsto\left(-x,-y\right) $. The basic invariant functions are
\begin{equation}
a=x^{2}\text{, }b=xy\text{, }c=y^{2},
\notag\end{equation}
they satisfy the constraint
\begin{equation}
ac=b^{2}
\notag\end{equation}
that determines a cone in the space $ \left(a,b,c\right) $. Therefore the quotient of the
plane by the action of $ {\mathbb Z}_{2} $ is a cone. \end{example}
\begin{example} \label{ex7.2}\myLabel{ex7.2}\relax The previous example is an (antisymmetrical) component
of the action of
$ {\mathbb Z}_{2}={\mathfrak S}_{2} $ on the product of the plane by itself by interchanging the factors,
so it suits the situation with the symmetrical power well.
Let $ M $ be a two-dimensional Poisson manifold. Suppose first that in a
vicinity of a given point the Poisson structure is nondegenerate. Then we
can choose local coordinates $ X $, $ Y $ such that the structure is
$ \frac{\partial}{\partial X}\wedge\frac{\partial}{\partial Y} $. On $ M\times M $ we can consider the coordinates $ X_{1} $, $ Y_{1} $, $ X_{2} $, $ Y_{2} $, or
$ \xi=\frac{1}{\sqrt{2}}\left(X_{1}+X_{2}\right) $, $ \eta=\frac{1}{\sqrt{2}}\left(Y_{1}+Y_{2}\right) $, $ x=\frac{1}{\sqrt{2}}\left(X_{1}-X_{2}\right) $, $ y=\frac{1}{\sqrt{2}}\left(Y_{1}-Y_{2}\right) $.
The cross-product Poisson structure can be written as
\begin{equation}
\frac{\partial}{\partial\xi}\wedge\frac{\partial}{\partial\eta}+\frac{\partial}{\partial x}\wedge\frac{\partial}{\partial y}.
\notag\end{equation}
The functions on $ M\times M/{\mathfrak S}_{2} $ are generated by $ \xi,\eta $, and $ a,b,c $ (as above), hence
the quotient is a product of a plane and a cone. Let us consider a blow-up
of this manifold in the singular stratum, i.e., in the product of the plane
and the vertex of the cone. Since all the structures (including the
Poisson) are cross-product structures, it is sufficient to consider the
blow-up of a cone in its vertex.
\end{example}
\begin{example} \label{ex7.3}\myLabel{ex7.3}\relax Let us consider in the situation of the example~\ref{ex7.1}
the Poisson structure $ \frac{\partial}{\partial x}\wedge\frac{\partial}{\partial y} $ on the plane $ \left(x,y\right) $. Since the Poisson
bracket of two $ {\mathbb Z}_{2} $-invariant functions is again $ {\mathbb Z}_{2} $-invariant, we can consider
the corresponding bivector field
on the smooth part of the quotient-cone $ K $. Let $ \widetilde K $ be a blow-up of this
cone in its vertex. Then on the open part of $ \widetilde K $ a bivector field is
defined. We claim that this bivector field extends to the whole $ \widetilde K $
without singularity, and that the corresponding Poisson structure on $ \widetilde K $
is non-degenerate.
Indeed, in a local coordinate frame $ \left(\alpha,\beta\right) $ on $ \widetilde K $, where $ \alpha=y/x $,
$ \beta=x^{2} $, the corresponding $ 2 $-form $ dx\wedge dy $ can be written as $ -\frac{1}{2}d\alpha\wedge d\beta $, therefore
\begin{equation}
\frac{\partial}{\partial x}\wedge\frac{\partial}{\partial y}=-2\frac{\partial}{\partial\alpha}\wedge\frac{\partial}{\partial\beta}.
\notag\end{equation}
\end{example}
\begin{remark} Therefore in the situation of the example~\ref{ex7.2} on the (smooth)
blow-up
of $ M\times M/{\mathfrak S}_{2} $ in the vertex a non-degenerate Poisson structure is defined.
If the original Poisson structure on $ M $ was degenerate, we can represent
it as a difference of two non-degenerate bivector fields. Both these
fields can be
raised to $ M\times M/{\mathfrak S}_{2} $ without singularity. Since the correspondence between
bivector fields on $ M $ and on $ M\times M/{\mathfrak S}_{2} $ is linear, the raising of the original
Poisson structures is a difference of two non-singular bivector fields,
and therefore is also non-singular. \end{remark}
\begin{definition} Let $ M $ be $ 2 $-dimensional manifold. Let us call the blow-up of
$ M\times M/{\mathfrak S}_{2} $ in the singular stratum {\em a symmetric square\/} $ S^{2}M $ {\em of\/} $ M $. The
above considerations show that if $ M $ is equipped
with a Poisson or symplectic structure, then $ S^{2}M $ is also equipped with a
Poisson or symplectic structure. \end{definition}
It is known that to define a ``good'' notion of a symmetrical power
(anywhere beyond the notion of the symmetrical square) is difficult.
However, in the case of symmetrical power of $ 2 $-dimensional
manifold the notion of {\em the Hilbert scheme\/} is sufficient in many cases.
Let us remind that a {\em desingularization\/} of a given variety $ X $ is a
manifold $ X' $ with a mapping $ \pi\colon X'\to X $ such that $ \pi $ is an isomorphism over
the open subset $ U $ of smooth points of $ X $. One of the ways to describe a
desingularization is to demonstrate an inclusion of $ U $ into some variety
in such a way that the closure $ X'=\overline U $ of the image of $ U $ is smooth. In this
case we should yet show the existence of the mapping $ X'\to X $, but usually it
is not difficult. We want to describe the Hilbert scheme of
a manifold $ M $ as a desingularization of the manifold $ \underbrace{M\times M\times\dots \times M}_{n\text{ times}}/{\mathfrak S}_{n} $.
\begin{theorem} Let us associate to an $ n $-tuple of different points on a
smooth two-dimensional manifold $ M $ a vector space $ I $ of functions that
vanish at
this points. Let $ V $ be the image of this map in the Grassmannian $ \operatorname{Gr} $ of
subspaces of codimension $ n $ in the ring $ A $ of functions on $ M $. Then
\begin{enumerate}
\item
The closure $ \overline V $ of $ V $ is a smooth subvariety of $ \operatorname{Gr} $;
\item
Points of the manifold $ \overline V $ are exactly ideals of codimension $ n $;
\item
The only component of codimension 1 of $ \overline V\smallsetminus V \subset \overline V $ consists of ideals
with support in $ n-1 $ points, i.e., to collision of only two points on $ M $;
\item
The tangent space to $ \overline V $ at an ideal $ I\in\overline V $ is $ \operatorname{Hom}_{A}\left(I,A/I\right)\subset\operatorname{Hom}_{{\mathbb C}}\left(I,A/I\right) $.
\item
If $ n=2 $, then $ \overline V $ is a blow-up of $ M\times M/{\mathfrak S}_{2} $ in the singular stratum.
\end{enumerate}
\end{theorem}
This manifold is called {\em a Hilbert scheme\/} of $ M $ (on the level $ n $). Now,
if $ M $ is equipped with a Poisson structure, then on the open subset $ V $ of
the Hilbert scheme a cross-product bivector field is defined. As we have
seen, this bivector field has no singularity on $ \overline V\smallsetminus V $ in the case $ n=2 $,
hence it has no singularity on the component of codimension 1 also in the
case of arbitrary $ n $. However, the Hartogs theorem claims that if a
function has no singularity outside of a subset of codimension $ >1 $, then
it has an extension without singularity. Therefore the Poisson structure
on $ V $ has an extension to the whole $ \overline V $.
If the Poisson structure on $ M $ is non-degenerate, then the same
discussion applied to the corresponding symplectic form shows that the
symplectic form can be extended to the whole $ \overline V $ without singularity.
Therefore the Poisson structure on $ \overline V $ is non-degenerate.
\begin{remark} The above definition of the Hilbert scheme is applicable only
in the case when the manifold $ M $ is affine, so the ring of functions on $ M $
is sufficiently reach. In the other case we should just consider
{\em subsheaves\/} $ {\mathcal I} $ of the sheaf $ {\mathcal O} $ instead of ideals in $ {\mathcal O}\left(M\right) $, and $ \dim \Gamma\left({\mathcal O}/{\mathcal I}\right) $
instead of $ \dim A/I $. However, we will abuse the notations and will work
with the Hilbert scheme as if it consists of ideals even in the
case of projective $ M $. \end{remark}
\begin{corollary} It is possible to define a notion of a {\em symmetric
power\/} $ S^{n}M $ of a
$ 2 $-dimensional Poisson manifold $ M $, that is also a Poisson manifold. To do
this it is
sufficient to consider the Hilbert scheme of $ M $. If the Poisson structure
on $ M $ is non-degenerate, the corresponding Poisson structure on $ S^{n}M $ is
non-degenerate too. Since the correspondence between these Poisson
structures is linear, the symmetrical power of a bihamiltonian manifold
is a bihamiltonian manifold. \end{corollary}
\subsection{A mapping to the symmetrical power }\label{s1.7}\myLabel{s1.7}\relax Now basing on a local
$ 2n $-dimensional bihamiltonian manifold $ M $ in a
vicinity
of a regular point we have constructed a bihamiltonian manifold $ S^{n}M^{\left(2\right)} $
and a mapping from the subset of good points on $ M $ into this bihamiltonian
manifold. This mapping preserves pairs of Poisson
structures. What we want
to do now is to show that we can extend this mapping to the whole $ M $.
In fact such an extension should associate an ideal of
codimension $ n $ on $ M^{\left(2\right)} $ to any point $ m\in M $. Let us construct this mapping on
the set of good points of $ M $. If we do all constructions algebraically,
then we will be able to apply them in the case of an arbitrary point of
$ M $.
\begin{lemma} Let $ m $ is a good point on $ M $. Let $ C\subset M\times M^{\left(2\right)} $ be the incidence set
consisting of pairs $ \left(m,L\right) $ such that $ m\in L $. Consider two projections $ \pi_{1} $ and
$ \pi_{2} $ from $ C $ to $ M $ and to $ M^{\left(2\right)} $. Let $ S $ be a set of weak leaves that pass
through $ m $,
\begin{equation}
S=\pi_{2}\pi_{1}^{-1}\left(\left\{m\right\}\right).
\notag\end{equation}
Let $ I_{m} $ be an ideal of functions on $ M $ vanishing at $ m $. Then the ideal
$ \pi_{2*}\pi_{1}^{*}\left(I_{m}\right) $ in the algebra of functions on $ M^{\left(2\right)} $ consists of vanishing on
$ S\subset M^{\left(2\right)} $ functions.
\end{lemma}
\begin{proof} Though this fact is absolutely standard in algebraic geometry,
we give here a proof.
We want to associate to a good point $ m $ of $ M $ an ideal on $ M^{\left(2\right)} $ with
zeros in the weak leaves passing through $ m $. We can ``pass objects on $ M $
through correspondence $ C $'': we can consider the inverse image with respect
to the first projection $ \pi_{1} $ (this gives us an object on $ C $) and after that
a direct image with respect to the second projection $ \pi_{2} $.
Now $ \pi_{1}^{*}I_{m} $ is the ideal generated by lifts of functions from the
ideal $ I_{m} $, i.e., by lifts the equations of the point $ m $. So if the point
$ m $ has equations $ z_{k}=0 $, were $ z_{k} $ are coordinates on $ M $, then this ideal is
generated by the functions $ z_{k} $ considered as functions on $ C $. If the point
$ m $ is good, then the projection $ \pi_{1} $ is locally a nonramified covering,
hence the ideal consists of functions that vanish on all $ n $ points in
$ \pi_{1}^{-1}\left(m\right) $.
The direct image of the ideal consists of functions such that
their inverse image is in the ideal. In our case if $ m $ is a good point,
then the corresponding functions on $ M^{\left(2\right)} $ should vanish at the points
$ \pi_{2}\pi_{1}^{-1}\left(m\right) $. \end{proof}
Therefore the described algorithm $ m\mapsto\pi_{2*}\pi_{1}^{*}\left(I_{m}\right) $ is indeed
what we need, at least at good points. Consider it at an arbitrary point
now. In fact we have constructed a mapping that to any point of $ M $
associates an ideal on $ M^{\left(2\right)} $. What we need to prove is the fact that to any
point of $ M $ we associate an ideal {\em of codimension\/} $ n $ {\em indeed}. That signifies
that the number of weak leaves passing through a given point of $ M $ (and
taken with proper multiplicities) does not depend on the point of $ M $ we
choose. On the algebraic language this is denoted by the words {\em the map is
flat}. So the only thing we need to do now is to prove that the projection
$ \pi_{1}\colon C\to M $ is flat.
\begin{theorem} \label{th1.5}\myLabel{th1.5}\relax If the mapping $ \pi_{1}\colon C\to M $ is flat over a neighborhood of
$ m\in M $,
then for $ m' $ in this neighborhood $ \operatorname{codim} \pi_{2*}\pi_{1}^{*}\left(I_{m'}\right)=n $, and the mapping
\begin{equation}
M\to S^{n}M^{\left(2\right)}\colon m'\mapsto\pi_{2*}\pi_{1}^{*}\left(I_{m'}\right)
\notag\end{equation}
is compatible with bihamiltonian structures. Both the Poisson structures
can be extended from an open subset of $ M^{\left(2\right)} $ to the whole space $ M^{\left(2\right)} $ if
$ M^{\left(2\right)} $ is smooth. Moreover, if one of the
Poisson structures on $ M $ is nondegenerate at $ m $, and the space $ M^{\left(2\right)} $ is
smooth, then this mapping is a
local isomorphism of bihamiltonian manifolds. Here we consider a
bihamiltonian structure on $ S^{n}M^{\left(2\right)} $ defined in the previous section. \end{theorem}
\begin{proof} The mapping
$ M\to S^{n}M^{\left(2\right)} $
preserves the Poisson structures on an open dense subset of good points of
$ M $. Therefore it preserves the Poisson brackets everywhere.
Suppose that $ M^{\left(2\right)} $ is smooth and a Poisson structure there has a
singularity on a curve $ L $. Then the corresponding $ 2 $-form has a zero on
this curve. However, it can have a singularity on some other curve $ L' $,
so consider a point on $ L\smallsetminus L' $. The discussion above shows
that the corresponding $ 2 $-form on an open subset of $ S^{k}M^{\left(2\right)} $ is
non-singular and
degenerate on a hypersurface. Consider a generic point on the intersection
of the image of $ M $ and the corresponding hypersurface in $ S^{n}M^{\left(2\right)} $. A
neighborhood of this point is a direct product of $ S^{k}M^{\left(2\right)} $ and $ S^{n-k}M^{\left(2\right)} $ for
an appropriate $ k $, and the $ 2 $-form is a direct product of a non-singular
form on $ S^{k}M^{\left(2\right)} $ and some (possibly singular) form on $ S^{n-k}M^{\left(2\right)} $. Consider
two functions on $ S^{k}M^{\left(2\right)} $ and corresponding functions on $ S^{n}M^{\left(2\right)} $. The
Poisson bracket of these functions has a pole on a hypersurface, but no
zero nearby. Therefore the Poisson bracket of the corresponding functions
on $ M $ is singular, what is impossible.
Now to prove that this mapping is a {\em local isomorphism\/} we should
only note that if one of the Poisson structures on the
bihamiltonian manifold $ M $ is nondegenerate, then by construction the
corresponding Poisson structure on the set of weak leaves $ M^{\left(2\right)} $ is also
nondegenerate, therefore the corresponding Poisson structure on $ S^{n}M^{\left(2\right)} $
is nondegenerate. Since the map $ M\to S^{n}M^{\left(2\right)} $ preserves the Poisson
structures, the Jacobian of this map is non-vanishing, therefore this map
is a local isomorphism.\end{proof}
\begin{remark} The previous theorem is adapted for a classification of
bihamiltonian structures in a neighborhood of a regular point, as in a
corollary below. However, it can be generalized a lot after introduction
of a new definition.
Let us call a closure of a weak leaf of codimension 2 a {\em generalized\/}
weak leaf, and let us extend this definition by taking a limit: call a
submanifold a generalized weak leaf if it can be approximated by closures
of weak leaves: \end{remark}
\begin{definition} A submanifold $ L_{0} $ of a bihamiltonian manifold $ M $ is called a
generalized weak leaf if there exists a locally close submanifold $ {\mathcal L}\subset M $
such that
in a neighborhood of any point there exists a function $ \psi\colon {\mathcal L}\to{\mathbb C} $ such that
$ \psi^{-1}\left(t\right)\buildrel{\text{def}}\over{=}L_{t}' $ is a closure of a weak leaf of codimension 2 if $ t\not=0 $ and is $ L_{0} $ if
$ t=0 $. \end{definition}
\begin{amplification}[The weak classification theorem] Let $ M^{\left(2\right)}{}' $ be a
set of generalized weak leaves and $ C' $ be the
corresponding incidence set:
\begin{equation}
C'=\left\{\left(m,L\right) \mid m\in L, L\text{ is a generalized weak leaf}\right\}\subset M\times M^{\left(2\right)}{}'.
\notag\end{equation}
Suppose that $ \pi_{1}\colon C'\to M $ is flat. Then the conclusions of the theorem
~\ref{th1.5} remain true, if we change $ M^{\left(2\right)} $ to $ M^{\left(2\right)}{}' $.
\end{amplification}
\begin{corollary}[The strong classification theorem] Let $ m $ be a regular point on a
bihamiltonian manifold $ M $ of
dimension $ 2n $. Suppose that any weak leaf on $ M $ intersects the set of good
points. Consider a (partial) mapping
\begin{equation}
M\to S^{n}M^{\left(2\right)}
\notag\end{equation}
defined on the set of good points. Then this mapping extends onto a whole
neighborhood of $ m $ and is a local isomorphism of bihamiltonian manifolds.
Therefore to any such manifold we associate a canonically defined
$ 2 $-dimensional bihamiltonian manifold $ M^{\left(2\right)} $, and we canonically identify
the initial manifold with the Hilbert scheme of this $ 2 $-dimensional manifold. \end{corollary}
The proof is already completed modulo the flatness result. As
usual, the
proofs of flatness of particular maps are absolutely straightforward and
a little bit dull. We postpone it until the appendix in the
section~\ref{h4}.
Let us consider the conditions of the weak classification theorem.
It is possible to construct an example of a bihamiltonian manifold with
non-smooth set of weak leaves (see the section~\ref{s6.8}). This shows that we
cannot drop the restriction of smoothness in the theorem. Moreover, this
example shows in fact that we cannot drop this condition even in the case
of the strong classification theorem. However, in the particular case of
this example the theorem remains true if we consider a {\em normalization\/} of
the Hilbert scheme instead of the Hilbert scheme itself. This shows that
there exist some potential for generalization of the theorem.
We cannot drop the condition of non-degeneracy either. Indeed,
consider a $ 2 $-dimensional bihamiltonian manifold such that the two
bivector fields have common zeros of second order. Then in some points on
the corresponding Hilbert scheme the bivector fields also have zeros of
the second order, therefore we can pull these bivector fields up to a
blow-up of this point. Now, if we consider this blow-up, we can see that
the mapping to the Hilbert scheme of the set of weak leaves
coincides with the mapping of this blow-up, therefore not an isomorphism.
\section{The odd-dimensional case }\label{h2}\myLabel{h2}\relax
We have seen that in the case of even dimension the set of weak
leaves is $ 2 $-dimensional and the original manifold can be canonically
reconstructed basing on this set. Let us try to proceed with this program
as far as we can in the odd-dimensional case.
\subsection{Facts from the linear algebra }\label{s2.1}\myLabel{s2.1}\relax First we want to give a more
vivid picture of a pair of bilinear forms in general position in an
odd-dimensional case. The theorem from the appendix gives us a good
picture in the case of pairs of mappings. In the section~\ref{s1.2} we have
already warned the reader that while the reading of the appendix in the
section~\ref{h3} was not nessecary, it was highly recommended. This warning is
still effective here, where we interpret what this theorem says in a
coordinate form. We strongly recommend to read the appendix on linear
algebra now, at least those concerning the Kroneker pairs and the
odd-dimensional case.
Let
us introduce a basis $ x_{l}=r_{1}^{l}r_{2}^{k-l} $ in the space $ S^{k}R $ (here $ R $ is spanned by
two vectors $ r_{1} $, $ r_{2} $) and a basis $ y_{l}=r_{1}^{l}r_{2}^{k-1-l} $ in the space $ S^{k-1}R $. Then
the two mappings $ \widetilde\alpha $, $ \widetilde\beta $ of the Kroneker pair $ K_{k}^{+} $ can be written as
\begin{equation}
y_{l}\buildrel{\widetilde\alpha}\over{\mapsto}x_{l+1},\quad y_{l}\buildrel{\widetilde\beta}\over{\mapsto}x_{l}.
\notag\end{equation}
However, for the Kroneker pair $ K_{k}^{-} $ we want to use a different
description. Let us consider a pair of {\em dual\/} mappings to the mappings $ K_{k}^{+} $.
It is clear that this pair is undecomposable, therefore isomorphic to the
pair $ K_{k}^{-} $ (see the theorem). In the dual basis it can be written as
\begin{equation}
x_{l}^{*}\buildrel{\widetilde\alpha^{*}}\over{\mapsto}y_{l-1}^{*},\quad x_{l}^{*}\buildrel{\widetilde\beta^{*}}\over{\mapsto}y_{l}^{*}.
\notag\end{equation}
Here we set $ y_{-1}^{*}=y_{k}^{*}=0 $.
That means that basing on the theorem~\ref{th3.1} we can describe a pair of
skew-symmetric forms $ \alpha $, $ \beta $ (in general position) in an
$ \left(2k+1\right) $-dimensional vector space $ V $ as following:
there is a basis
$ \left(x_{0},\dots ,x_{k},y_{0}^{*},\dots ,y_{k-1}^{*}\right) $ in this space and the only non-vanishing basic
pairings are
\begin{equation}
\alpha\left(x_{i+1},y_{i}^{*}\right)=1,\quad i=0,\dots ,k-1
\notag\end{equation}
and
\begin{equation}
\beta\left(x_{i},y_{i}^{*}\right)=1,\quad i=0,\dots ,k-1,
\notag\end{equation}
(so if $ k=0 $ both pairings (in $ 1 $-dimensional!) space with the basis $ x_{0} $
vanish). The kernel of the combination $ \alpha-\lambda\beta $ is spanned by the
vector $ x_{0}+\lambda x_{1}+\lambda^{2}x_{2}+\dots +\lambda^{k}x_{k} $. In accordance with what the theorem says,
these kernels evidently span the space $ W_{1} $ generated by $ \left(x_{i}\right) $, $ i=0,\dots ,k $, and
form in the projectivization of this space an image of the {\em Veronese
inclusion\/}: $ {\mathbb P}^{1}\to{\mathbb P}^{k}\colon \left(1:\lambda\right)\mapsto\left(1:\lambda:\lambda^{2}:\dots :\lambda^{k}\right) $.
\subsection{The space of weak leaves }\label{s2.2}\myLabel{s2.2}\relax Let us consider now a (local)
$ \left(2k+1\right) $-dimensional bihamiltonian manifold
$ M $ and the space $ M^{\left(2\right)} $ of weak leaves in it.\footnote{Let us remind that a weak leaf is a symplectic leaf of some linear
combination of two Poisson structures.} Suppose again that {\em the values
of bivector fields\/} $ \eta_{1}|_{m} $, $ \eta_{2}|_{m} $ at the given point $ m\in M $ are in general
position (then they are in general position also in some neighborhood of
$ m $). First of all we want to show that we don't misuse the notation here:
\begin{lemma} The space $ M^{\left(2\right)} $ is $ 2 $-dimensional. \end{lemma}
\begin{proof} Indeed, consider again the incidence set $ C\subset M\times M^{\left(2\right)} $ of pairs
$ \left(x,L\right) $, $ x\in L $. The dimension of this set is
\begin{equation}
\dim C=\dim M^{\left(2\right)}+\dim L =\dim M+\delta,
\notag\end{equation}
where $ \delta $ is the dimension of the set of weak leaves containing a given
point $ m\in M $. However, in the odd-dimensional case {\em any linear combination of
the bilinear forms\/} has a kernel, and this kernel is $ 1 $-dimensional (in the
case of general position). Hence $ \dim L=\dim M-1 $, $ \delta $ is the dimension of the
Veronese curve, i.e., $ \delta=1 $. Hence $ \dim M^{\left(2\right)}=2 $.\end{proof}
So we see a big contrast with the even-dimensional case. A whole
$ 1 $-parametric family of weak leaves is passing through a given point. That
means that to a given point corresponds a (rational) curve on the surface
$ M^{\left(2\right)} $.
However, the greatest difference with the even-dimensional case is
that {\em the kernels do not span the whole cotangent space at the given
point}, but a subspace of codimension $ k $. (Here again we consider a bivector
as a bilinear form in the
cotangent space.) From the other side, the kernel is a normal space to
the corresponding symplectic leaf, therefore
the intersection of all the weak leaves passing through a
given point is not that point, but a submanifold of dimension $ k $ passing
through this point. Indeed, the sum of normal spaces to a family of
subspaces is the normal space to the intersection of this family. This
means that the intersection of the {\em tangent spaces\/} to weak leaves passing
through $ m $ is $ k $-dimensional. Now we need to prove that this is true not
only on the level of tangent spaces, but in a neighborhood of $ m $.
To do this we can note that it is sufficient to consider a sum of $ k $
kernels corresponding to $ k+1 $ different values $ \left(\lambda_{0},\dots ,\lambda_{k}\right) $ of $ \lambda $, since
this sum
coincides with the whole subspace $ W_{1} $ spanned by all the kernels. Therefore the
tangent space to the
(evidently transversal) intersection $ L_{m,\lambda_{0},\dots ,\lambda_{k}} $ of $ k+1 $ corresponding
weak leaves passing through $ m $ coincides with the
tangent space to the intersection of all the weak leaves passing through
$ m $. Now {\em integration\/} gives us that {\em any\/} weak leaf passing through $ m $ contains
$ L_{m,\lambda_{0},\dots ,\lambda_{k}} $. Submanifolds $ L_{m',\lambda_{0},\dots ,\lambda_{k}} $ for different points $ m' $ form a
foliation of dimension $ k $ on $ M $. We can conclude that any weak leaf that
intersects some leaf of this foliation should contain it. It is clear
that this foliation does not depend on the particular values of
$ \left(\lambda_{0},\dots ,\lambda_{k}\right) $. Let us call this foliation $ {\mathcal L} $, and call a leaf of this
foliation $ {\mathcal L}_{m}= L_{m,\lambda_{0},\dots ,\lambda_{k}} $.
\begin{corollary} To any point $ m\in M $ corresponds a rational curve on the
space of weak leaves $ M^{\left(2\right)} $. Points on the same leaf of $ {\mathcal L} $ correspond to the
same rational curve on $ M^{\left(2\right)} $, points on different leaves correspond to
different curves. \end{corollary}
\subsection{Veronese webs }\label{s2.3}\myLabel{s2.3}\relax We see that in contrast with the even-dimensional
case the natural correspondence between $ M $ and $ M^{\left(2\right)} $ glues together points on a
leaf of the foliation $ {\mathcal L} $. Therefore
we cannot directly reconstruct $ M $
basing on $ M^{\left(2\right)} $, but only the {\em local base\/} of the foliation $ {\mathcal L} $. Let us
call
this $ \left(k+1\right) $-dimensional base $ X_{M} $. Here we want to describe some geometrical
structure on this manifold. We will be able to construct $ M^{\left(2\right)} $ basing on
this structure alone. Moreover, this base and this structure on it can
be canonically
reconstructed basing on $ M^{\left(2\right)} $. After that the correspondence
\begin{equation}
\left(M,\eta_{1},\eta_{2}\right)\mapsto M^{\left(2\right)}
\notag\end{equation}
can be passed via $ X_{M} $:
\begin{equation}
\left(M,\eta_{1},\eta_{2}\right)\mapsto X_{M}\mapsto M^{\left(2\right)},
\notag\end{equation}
and the natural correspondence between $ X_{M} $ and $ M^{\left(2\right)} $ does
not glue any two points on $ X_{M}. $\footnote{In fact it is possible to reconstruct $ \left(M,\eta_{1},\eta_{2}\right) $ itself basing
on $ X_{M} $ (at
least locally), however not canonically but only up to a (local)
diffeomorphism (see \cite{GelZakh91Web}). We show how to reconstruct $ M $
(without Poisson structures) in the section~\ref{s2.4}.}
Since any weak leaf (of codimension 1) either contains a leaf of
$ {\mathcal L} $, or does not
intersect it, it corresponds to a submanifold of codimension 1 of $ X_{M} $. For
a fixed $ \lambda $ the symplectic leaves of $ \alpha-\lambda\beta $ (of codimension 1) form a
foliation on $ M $, that can be pushed down to a foliation on $ X_{M} $. Hence we
have a parameterized
by $ \lambda\in{\mathbb P}^{1} $ family of foliations on $ X_{M} $. For a given point $ x\in X_{M} $ to any $ \lambda\in{\mathbb P}^{1} $
we can associate the normal subspace to the
passing through $ x
$ leaf of the foliation with parameter $ \lambda $. We can consider this
subspace as a point in the projectivization $ PT_{x}^{*}X_{M} $ of the cotangent
subspace at $ x $. The results above show that this mapping $ {\mathbb P}^{1}\to PT_{x}^{*}X_{M} $:
\begin{equation}
\lambda\mapsto\text{ a normal space to the projection of the symplectic leaf
for }\alpha-\lambda\beta
\notag\end{equation}
is (in an appropriate coordinate system) isomorphic to the {\em Veronese
inclusion\/}
$ \lambda\mapsto\left(1:\lambda:\dots :\lambda^{k}\right) $.
Such an object has so beautiful geometry that it is worthy a name.
\begin{definition} A {\em Veronese curve\/} is an inclusion of $ {\mathbb P}^{1} $ into a projective
space isomorphic to a Veronese inclusion $ {\mathbb P}^{1}\hookrightarrow{\mathbb P}^{k} $. \end{definition}
\begin{definition} A {\em Veronese web\/} is a $ \left(k+1\right) $-dimensional manifold $ X $ with a
parameterized by $ \lambda\in{\mathbb P}^{1} $ family of foliations $ \left\{{\mathcal F}_{\lambda}\right\} $ of codimension 1 on $ X $ such
that given a point $ x\in X $ the normal lines $ N_{x}{\mathcal F}_{\lambda,x}\subset T_{x}^{*}X $ to the leaves $ {\mathcal F}_{\lambda,x} $
of foliations passing through $ x $ form a {\em parameterized by\/} $ \lambda\in{\mathbb P}^{1} $ {\em Veronese
curve\/}
\begin{equation}
\lambda\mapsto N_{x}{\mathcal F}_{\lambda,x}
\notag\end{equation}
in $ PT_{x}^{*}X $. \end{definition}
\subsection{Reconstruction of the bihamiltonian manifold basing on a Veronese web
}\label{s2.4}\myLabel{s2.4}\relax Now we can (canonically) associate
a Veronese web $ X_{M}
$ to any odd-dimensional bihamiltonian system $ M $ in general position. We
say that this bihamiltonian
manifold is a {\em bihamiltonian structure over\/} $ X_{M} $. A remarkable
fact is that {\em this correspondence is invertible up to a diffeomorphism\/}:
\begin{theorem}[\cite{GelZakh91Web}] Let $ X $ be a (local) Veronese web. Basing on $ X $
we can construct a
bihamiltonian manifold $ M_{X} $ with a natural projection to $ X $. The Veronese
web $ X_{M_{X}} $ constructed basing on $ M_{X} $ is naturally isomorphic to $ X $.
If we consider analytic manifolds and if the Veronese web $ X $
corresponds in the described above way to a bihamiltonian manifold $ M $,
i.e., $ X=X_{M} $, then the bihamiltonian manifold $ M_{X}=M_{X_{M}} $ is locally isomorphic
to $ M $ and the map $ M\to X_{M} $ corresponds under this isomorphism to the map
$ M_{X_{M}}\to X_{M} $ (however, this isomorphism is not canonical). \end{theorem}
Here we do not want to discuss a proof of this theorem, however, we
want to explain briefly the construction of the bihamiltonian manifold $ M_{X} $
as a plain manifold (i.e., we cannot explain here the construction of two
Poisson structures on $ M_{X} $). Consider the cotangent bundle $ T^{*}X $. We
have a Veronese inclusion of the same projective line $ {\mathbb P}^{1} $ in the
projectivization of any
vector space of this bundle. Let us consider the Veronese inclusion $ {\mathbb P}^{1}\to{\mathbb P}^{k} $.
It is easy to see that for any projective
transformation of $ {\mathbb P}^{1} $ we can find a (unique) projective transformation
of $ {\mathbb P}^{k} $
that in the restriction to the image of $ {\mathbb P}^{1} $ gives the given transformation
of $ {\mathbb P}^{1} $.
Hence the same is true for any Veronese curve, in particular for any
point $ x\in X $.
Let us denote by $ {\mathcal S} $ the $ 2 $-dimensional coordinate vector space (so $ P{\mathcal S}={\mathbb P}^{1} $).\footnote{If the Veronese web is associated with a bihamiltonian manifold,
we can identify $ {\mathcal S} $ with the space of linear combinations of two Poisson
structures.}
Hence with any volume-preserving linear transformation of $ {\mathcal S} $
(i.e., an element of $ \operatorname{SL}_{2}=\operatorname{SL}\left({\mathcal S}\right) $)
we can
associate a volume-preserving transformation
of $ T_{x}^{*}X $ (i.e., an element of $ \operatorname{SL}\left(T_{x}^{*}X\right) $). Therefore we have an $ \operatorname{SL}_{2} $-structure
on the cotangent bundle of $ X $. Now we are going to do the following (usual
in the theory of vector bundles) trick: to any vector bundle with an
action of a group we can associate a {\em principal bundle\/} for this group
over $ X $,
and to any representation of this group we can associate another vector
bundle over $ X $. It is easy to see that the cotangent bundle on $ X $
considered as an $ \operatorname{SL}_{2} $-bundle corresponds in this
consideration to the representation of $ \operatorname{SL}\left({\mathcal S}\right) $ in the $ k $-th symmetrical
power $ S^{k}{\mathcal S} $ of $ {\mathcal S} $.
Now we can define $ M_{X} $ as a total space of the vector bundle
corresponding to the {\em previous\/} symmetrical power $ S^{k-1}{\mathcal S} $. Since this
argument is a little bit misleading, we want to give a more direct
definition. Let us consider the vector bundle $ T^{*}X\otimes{\mathcal S} $ over $ X $. The action of
the group $ \operatorname{SL}\left({\mathcal S}\right) $ on the fibers of this bundle decomposes canonically into
a direct sum of two representations: one $ \left(k+2\right) $-dimensional, another
$ k $-dimensional. Now $ M_{X} $ is a total space of the vector bundle over $ X $
corresponding to the second component with respect to the action of $ \operatorname{SL}_{2} $.
\begin{definition} We call $ M_{X} $ a {\em subcotangent bundle\/} for $ X $ and denote it by
$ T^{*\left(-1\right)}X $. \end{definition}
In the same way as one can define a Poisson structure on the cotangent
bundle to a manifold, it is possible to define a family of Poisson
structures on the subcotangent bundle parameterized by the vector space $ {\mathcal S} $,
i.e., a bihamiltonian structure. However, this definition in the present
form \cite{GelZakh91Web} is rather ugly. The situation is very similar to a
try to
define a Poisson structure on the cotangent bundle without a reference to
the symplectic structure\footnote{I.e., without the use of the operation of inversion of a matrix.} on this manifold: it is possible, but we do not
know a ``direct'' way to do it.
\subsection{The double complex }\label{s2.5}\myLabel{s2.5}\relax Another remaining question is why if we start
with a bihamiltonian manifold $ M $,
construct a Veronese web $ X_{M} $ basing on it and construct the bihamiltonian
structure $ T^{*\left(-1\right)}X_{M} $ basing on this web, we get two locally isomorphic
bihamiltonian structures $ M $ and $ T^{*\left(-1\right)}X_{M} $. Here we also do not know a direct
proof, in fact, in the $ C^{\infty} $-case we even do not know if this is true.
However, the algebraic formalism involved in the proof is so exotic
that we want to provide some details of this proof here (risking to annoy
the reader with an absence of precise definitions).
The proof is
based on the consideration of an analogue of the de Rham complex on
$ X $. This complex is
associated with the vector bundle $ T^{*\left(-1\right)}X\to X $ instead of $ T^{*}X\to X $, i.e., it
is the
complex of sections of $ \Lambda^{\bullet}T^{*\left(-1\right)}X $. In the same way as it is possible to
define a differential of degree 1 on $ \Omega^{\bullet}=\Gamma\left(\Lambda^{\bullet}T^{*}X\right) $, we can define {\em two\/}
differentials $ d_{1} $, $ d_{2} $ on
\begin{equation}
\widetilde\Omega^{\bullet}=\Gamma\left(\Lambda^{\bullet}T^{*\left(-1\right)}X\right),
\notag\end{equation}
any linear combination of which is again a differential. The last
condition means
\begin{equation}
d_{1}^{2}=d_{2}^{2}=d_{1}d_{2}+d_{2}d_{1}=0.
\notag\end{equation}
(The only difference of this situation and of the definition of a
bicomplex is that we have only $ {\mathbb Z} $-grading, but not $ {\mathbb Z}^{2} $-grading.)
It is possible to show that any section $ \varphi $ of $ T^{*\left(-1\right)}X=\widetilde\Omega^{1} $ satisfying
\begin{equation}
d_{1}d_{2}\varphi=0\in\widetilde\Omega^{3}
\notag\end{equation}
gives rise to some bihamiltonian manifold $ M_{X,\varphi} $ over $ X $. Any bihamiltonian
manifold over $ X $ can be obtained in this way.
However, if $ \varphi_{1} $ and $ \varphi_{2} $ satisfy the above differential equation and
\begin{equation}
\varphi_{1}-\varphi_{2}=d_{1}\psi_{1}+d_{2}\psi_{2}\in\widetilde\Omega^{1},\quad \varphi_{1,2}\in\widetilde\Omega^{0}=\Gamma\left({\mathcal O}\left(X\right)\right),
\notag\end{equation}
we can construct an isomorphism $ M_{x,\varphi_{1}}\to M_{X,\varphi_{2}} $ over $ X $ and visa versa.
Therefore the local
isomorphism classes of bihamiltonian structures over $ X $ correspond to
{\em double cohomology\/} classes
\begin{equation}
{\mathbb H}^{1}\widetilde\Omega^{\bullet}=\frac{\operatorname{Ker} d_{1}d_{2}\colon \widetilde\Omega^{1}\to\widetilde\Omega^{3}}{\left(\operatorname{Im} d_{1}\colon \widetilde\Omega^{0}\to\widetilde\Omega^{1}\right)+\left(\operatorname{Im} d_{2}\colon \widetilde\Omega^{0}\to\widetilde\Omega^{1}\right)}.
\notag\end{equation}
At least in the local holomorphic case we can prove that any space of {\em double
cohomology\/}
$ {\mathbb H}^{i}\widetilde\Omega^{\bullet} $, $ i\geq1 $, vanishes, and $ {\mathbb H}^{0}\widetilde\Omega^{\bullet}={\mathbb C} $. That finishes our sketch of the proof
of the theorem.
\subsection{The Kodaira theorem and Veronese webs }\label{s2.6}\myLabel{s2.6}\relax We have ``shown'' the
relation between odd-dimensional bihamiltonian
manifolds in general position and Veronese webs. However, we began this
discussion in connection with the question: can we reconstruct the
bihamiltonian structure basing on the set of weak leaves $ M^{\left(2\right)} $, i.e., in the
same spirit as when dealing with even-dimensional structures. It is clear
that we cannot hope to reconstruct more information than one contained
in the corresponding Veronese web, since the mapping $ M\mapsto M^{\left(2\right)} $ goes
via the Veronese web (since any weak leaf on $ M $ corresponds to a leaf
of some foliation on $ M^{\left(2\right)} $).
So let us call any leaf of any marked foliation on the Veronese web
$ X $ {\em a
weak leaf on\/} $ X $. Call the set of weak leaves $ X^{\left(2\right)} $ (The legality of this
notation is guaranteed by the isomorphism $ M^{\left(2\right)}\simeq X^{\left(2\right)} $ if $ X $ is associated
with a bihamiltonian manifold $ M $.) There is a natural
mapping $ \pi\colon X^{\left(2\right)}\to{\mathbb P}^{1} $, that send any leaf of the foliation $ {\mathcal F}_{\lambda} $ to $ \lambda $. To any
point $ x\in X $ there corresponds a section $ \Gamma_{x} $ of this bundle: to any $ \lambda $ we can
associate the leaf of the foliation $ {\mathcal F}_{\lambda} $ passing through $ x $.
However, in the holomorphic case we can indeed reconstruct the
Veronese web $ X $ basing on the set $ X^{\left(2\right)} $. In fact in contrast with the
even-dimensional case where we needed the bihamiltonian structure on $ M^{\left(2\right)} $ we
do not need any additional information here:
\begin{theorem} In the case of analytic manifolds if $ X $ is a (local) Veronese
web, then any
section of the projection $ X^{\left(2\right)}\to{\mathbb P}^{1} $ corresponds to a point on $ X $. \end{theorem}
\begin{proof} Let us note that in
the case when
$ X $ is a germ in neighborhood of $ x\in X $ the $ 2 $-dimensional manifold $ X^{\left(2\right)} $ is a
germ in a neighborhood of the curve $ \Gamma_{x}\subset X^{\left(2\right)} $.
Since we are working with germs of manifolds, it is sufficient to show
that the dimension of the set of section of the projection $ X^{\left(2\right)}\to{\mathbb P}^{1} $ that are
deformations of the curve $ \Gamma_{x} $ is equal to the dimension of $ X $. The Kodaira
theorem \cite{Kod} says that it is sufficient to show that {\em the degree\/} of the
normal
bundle to the (rational) curve $ \Gamma_{x} $ equals $ \dim X-1 $. Fix a point $ \left(x,\lambda\right) $ on
$ \Gamma_{x} $. The normal space $ N_{\left(x,\lambda\right)}\Gamma_{x} $ coincides with the normal space to the
leaf $ {\mathcal F}_{\lambda,x} $ of the foliation $ {\mathcal F}_{\lambda} $ at $ x $. Therefore {\em the normal bundle for\/} $ \Gamma_{x} $
{\em coincides with the tautological bundle for the Veronese inclusion\/}
\begin{equation}
\lambda\mapsto N_{x}{\mathcal F}_{\lambda,x},
\notag\end{equation}
and the degree of this bundle can be computed without any difficulty. \end{proof}
We came to the following
\begin{nwthrmi} Consider a germ of an (analytic) surface $ {\mathcal X} $ in a neighborhood
of a
rational curve $ \Gamma\subset{\mathcal X} $. Suppose that the degree of the normal bundle to $ \Gamma $ is
$ k\geq0 $. Then by the Kodaira theorem the curve $ \Gamma $ can be included in the
maximal ($ \left(k+1\right) $-parametric) family of rational curves on $ {\mathcal X} $
parameterized by some
$ \left(k+1\right) $-dimensional complex manifold $ X $. (Since $ {\mathcal X} $ is a germ only, $ X $ is a germ in
neighborhood of $ \Gamma\in X $.)
Fix a mapping $ \pi\colon {\mathcal X}\to{\mathbb P}^{1} $. Let us define the foliations $ {\mathcal F}_{\lambda} $, $ \lambda\in{\mathbb P}^{1} $. A
leaf of the foliation $ {\mathcal F}_{\lambda} $ on $ X $ consists of those curves on $ {\mathcal X} $ (i.e., points
of $ X $) that pass through a fixed point on $ \pi^{-1}\left(\lambda\right)\subset{\mathcal X} $. \end{nwthrmi}
\begin{theorem} Any mapping $ \pi\colon {\mathcal X}\to{\mathbb P}^{1} $ that is an isomorphism on $ \Gamma $ corresponds
in the specified above way to a canonical structure of a Veronese web on
$ X $. \end{theorem}
\begin{proof} The manifold $ X $ with a family of foliations $ {\mathcal F}_{\lambda} $ is already
defined in the theorem. What we need to do is to show that at any given
point $ x\in X $ the normal lines to the fibers of the foliations form a Veronese
curve. However, {\em any curve that is a deformation of a Veronese curve is
again a Veronese curve}, since any smooth curve of degree $ k $ in $ {\mathbb P}^{k} $ that
spans
the whole space $ {\mathbb P}^{k} $ is a Veronese curve. Therefore it is sufficient to
show that in one fixed point $ x $ of $ X $ this curve (corresponding to the
inclusion $ i\colon {\mathbb P}^{1}\hookrightarrow PT_{x}^{*}X $) is a Veronese curve, i.e., it spans the whole
space $ PT_{x}^{*}X $ and has degree $ k $. Of course, we choose the point $ \Gamma\in X $ as $ x $.
However, a generic curve on $ {\mathcal X} $ that is a small deformation of $ \Gamma $
intersects $ \Gamma $ in $ k $ points, therefore a generic tangent to $ X $ vector at $ x $
lies in a tangent space to a leaf of $ {\mathcal F}_{\lambda} $ for $ k $ different values of
$ \lambda $. Hence a generic hyperplane intersects the image of inclusion $ i:
{\mathbb P}^{1}\hookrightarrow PT_{x}^{*}X $ in $ k $ points.
If this image is contained in a hyperplane, then tangent spaces to
leaves of $ {\mathcal F}_{\lambda} $ contain a common vector $ v $. However, this vector corresponds to
an infinitesimal deformation of $ \Gamma $ that intersects $ \Gamma $ in any point of $ \Gamma $,
therefore to a 0 section of a normal bundle to $ \Gamma $. By the Kodaira theorem
the tangent space to $ X $ at $ x $ is {\em identified\/} with the space of sections of
this normal bundle, therefore $ v=0 $.\end{proof}
\subsection{Assembling the pieces of the puzzle }\label{s2.7}\myLabel{s2.7}\relax Now the correspondence
\begin{equation}
M\buildrel{\alpha}\over{\mapsto}M^{\left(2\right)}
\notag\end{equation}
(between odd-dimensional bihamiltonian manifolds and $ 2 $-dimensional
manifolds with a projection onto $ {\mathbb P}^{1} $ and a section of this projection) can be
broken into a chain
\begin{equation}
M\buildrel{\alpha_{1}}\over{\mapsto}X_{M}\buildrel{\alpha_{2}}\over{\mapsto}\left(X_{M}\right)^{\left(2\right)}=M^{\left(2\right)},
\notag\end{equation}
where the object in the middle is a Veronese web. The previous theorem
says that in the local case the last arrow can be
canonically inverted (i.e., inverted up to a canonically defined
isomorphism), so it is an equivalence of corresponding categories.
However, even in the local case the first arrow can be canonically
inverted only from the right, i.e., although the mapping
\begin{equation}
X\buildrel{\beta_{1}}\over{\mapsto}T^{*\left(-1\right)}X
\notag\end{equation}
satisfies the relation $ \beta_{1}\alpha_{1}\left(M\right)\simeq M $, $ \alpha_{1}\beta_{1}\left(X\right)\simeq X $, only the latter isomorphism
can be chosen canonically, i.e., compatibly with isomorphisms.
We can explain it in a more concrete way:
a bihamiltonian manifold has much more automorphisms than the corresponding
Veronese web. In fact, the set of automorphisms of a bihamiltonian
manifold that commute with the mapping\footnote{I.e., automorphisms of the bihamiltonian manifold that induce the
trivial automorphism of a Veronese web.} $ M\to X_{M} $ is similar to the described
above (trivial) classification of bihamiltonian manifolds over a Veronese
web:
it coincides with the vector space
\begin{equation}
\left(\operatorname{Im} d_{1}\colon \widetilde\Omega^{0}\to\widetilde\Omega^{1}\right)\cap\left(\operatorname{Im} d_{2}\colon \widetilde\Omega^{0}\to\widetilde\Omega^{1}\right)=\operatorname{Ker} d_{1}\oplus d_{2}\colon \widetilde\Omega^{1}\to\widetilde\Omega^{2}\oplus\widetilde\Omega^{2}.
\notag\end{equation}
It is possible to show that the number of such automorphisms coincides
with a $ k-1 $ times the number of functions of two variables. We consider a
particular case $ k=2 $ in the section~\ref{s2.9}.
\begin{remark} Now we can see that to give a local classification of
bihamiltonian manifolds it is sufficient to describe $ 1 $-{\em dimensional
nonlinear bundles\/} $ X^{\left(2\right)} $ over $ {\mathbb P}^{1} $ of a given degree. Here a degree $ \deg
X^{\left(2\right)} $ of a bundle
denotes the degree of its linearization: to have a degree a bundle
should have a section $ \Gamma $, and the vertical tangent bundle to this bundle at
this section should be of the given degree. Moreover, we should consider
only local objects of this sort, i.e., only germs of $ 2 $-dimensional
manifolds in a neighborhood of the section $ \Gamma $. A simple calculation shows
that the set of isomorphism classes of such objects is
parameterized (essentially---compare the remark~\ref{rem111} below) by
$ \deg X^{\left(2\right)} $ germs (at 0) of holomorphic functions of two variables.
Therefore to describe a Veronese web of dimension $ k $ up to an isomorphism we
need to provide $ k-1 $ functions of two variables. \end{remark}
\begin{remark} What we get in the previous remark is in fact a remarkable
fact: {\em the dimension of the parameter space for isomorphism classes of
odd-dimensional bihamiltonian manifolds (almost) does not depend on the
dimension of these manifolds!\/} The total amount of $ 5 $-dimensional
bihamiltonian manifolds is equal to the total amount of pairs of
$ 3 $-dimensional manifolds and so on! Indeed, in the $ 5 $-dimensional case the
space of parameters is a pair of functions of two variables, that is
twice as much as in the $ 3 $-dimensional case.
Moreover, it is possible to show that
in the holomorphic case there is an operation that associates a (local)
$ k $-dimensional Veronese web to a given (local) $ \left(k-1\right) $-dimensional Veronese
web and a (local) $ 2 $-dimensional Veronese web. Any $ k $-dimensional Veronese
web can be uniquely obtained in this way. So eventually we can obtain any
given Veronese web using this operation and starting from $ 2 $-dimensional
Veronese webs. (Unfortunately, we do not have a place here for a
discussion of this beautiful construction.) Note that this is compatible
with the calculation of the size of of the moduli space for Veronese
webs: to get $ k $-dimensional Veronese web we should fuse $ k-1 $ Veronese webs
of dimension 2, and to describe these webs we need $ k-1 $ functions
of two variable. \end{remark}
\subsection{$ 2 $-dimensional webs }\label{s2.8}\myLabel{s2.8}\relax Another remarkable fact is the possibility to
simplify a lot
the definition of $ 2 $-dimensional Veronese webs.
\begin{lemma} A $ 2 $-dimensional Veronese web is uniquely determined by any
three
different foliations from the $ {\mathbb P}^{1} $-parameterized family. Moreover, any
three
foliations on a surface such that the three tangent lines at any point are
different correspond to some Veronese web. \end{lemma}
The proof is trivial, since a Veronese inclusion $ {\mathbb P}^{1}\to{\mathbb P}^{1} $ is just an
isomorphism, that is uniquely determined by the image of any three
different points.
Therefore any three families of curves on a surface in general
position determine a Veronese web. This is a reason why we use the name
{\em web\/} here, since a web on a plane is exactly three families of curves in
general position. So to any web on a plane we can associate some $ 3 $-dimensional
bihamiltonian manifold. However, the
given above description of this manifold can be
simplified a lot in this particular case.
\subsection{$ 3 $-dimensional bihamiltonian structures }\label{s2.9}\myLabel{s2.9}\relax Let us consider a
$ 3 $-dimensional bihamiltonian manifold $ M $ with a mapping $ M\xrightarrow[]{\pi}X_{M} $. In this case
the Veronese web $ X_{M} $ is $ 2 $-dimensional, therefore we can consider instead
of $ {\mathbb P}^{1} $-parameterized family of foliations only three foliations corresponding
to values $ \lambda_{1},\lambda_{2},\lambda_{3}\in{\mathbb P}^{1} $. These foliations can be defined as level lines of
three functions
\begin{equation}
f_{1}=\operatorname{const}\text{, }f_{2}=\operatorname{const}\text{, }f_{3}=\operatorname{const}.
\notag\end{equation}
We can suppose that $ \lambda_{1}=\left(1:0\right) $, $ \lambda_{2}=\left(0:1\right) $, $ \lambda_{3}=\left(1:1\right) $. Then the functions
$ f_{i}\circ\pi $ are the Casimir functions\footnote{I.e., say, $ \left\{f_{1},g\right\}_{1}=0 $ for any function $ g $ on $ M $.} on $ M $ respective to the Poisson structures
$ \left\{,\right\}_{1} $, $ \left\{,\right\}_{2} $ and $ \left\{,\right\}_{1}+\left\{,\right\}_{2} $ correspondingly.
Locally we can represent $ M $ as a product of $ X_{M} $ and a line. Choose a
coordinate $ z $ along this line. We can choose functions $ x=f_{1} $ and $ y=f_{2} $ as
two coordinates on $ X_{M} $ and write $ f_{3}=F\left(x,y\right) $. Since the function $ x $ is a Casimir
function with
respect to the bracket $ \left\{,\right\}_{1} $, the bivector field that corresponds to that
bracket can be written as $ \varphi_{1}\left(x,y,z\right) \frac{\partial}{\partial y}\wedge\frac{\partial}{\partial z} $, $ \varphi_{1}\not=0 $. In the same
way the
second bivector field can be written as $ \varphi_{2}\left(x,y,z\right) \frac{\partial}{\partial x}\wedge\frac{\partial}{\partial z} $,
$ \varphi_{2}\not=0 $. Now the
condition that the function $ f_{3}=F\left(x,y\right) $ is a Casimir function with respect
to the bracket $ \left\{,\right\}_{1}+\left\{,\right\}_{2} $ gives us
\begin{equation}
\left( \varphi_{1}\left(x,y,z\right) \frac{\partial}{\partial y} + \varphi_{2}\left(x,y,z\right) \frac{\partial}{\partial x} \right) F\left(x,y\right)=0,
\notag\end{equation}
or
\begin{equation}
\frac{\varphi_{1}}{\varphi_{2}}=-\frac{F_{x}}{F_{y}}.
\notag\end{equation}
We have yet an arbitrariness in a choice of a function $ z $. It is easy to
understand (this is a variant of the d'Harboux theorem)
that by a change of the function $ z $ we can change $ \varphi_{1} $ in an arbitrary way.
In particular, we can choose $ \varphi_{1}=-F_{x} $, so $ \varphi_{2}=F_{y} $. So to a web
\begin{equation}
x=\operatorname{const}\text{, }y=\operatorname{const}\text{, }F\left(x,y\right)=\operatorname{const}
\notag\end{equation}
we associate a bihamiltonian manifold with coordinates $ \left(x,y,z\right) $ and two
Poisson brackets
\begin{equation}
\begin{aligned}
\left\{f\left(x,y,z\right),g\left(x,y,z\right)\right\}_{1} & =-F_{x}f_{y}g_{z}+F_{x}f_{z}g_{y},
\\
\left\{f\left(x,y,z\right),g\left(x,y,z\right)\right\}_{2} & =F_{y}f_{x}g_{z}-F_{y}f_{z}g_{x}.
\end{aligned}
\notag\end{equation}
\begin{remark} \label{rem111}\myLabel{rem111}\relax It is easy to see that in the example above we associated to a
$ 2 $-dimensional Veronese web a function $ F\left(x,y\right) $. Let us find the
arbitrariness in the definition of this function.
It is easy to see that
the function $ F $ is
defined up to a change of the form $ F_{1}=\varphi\left(F\right) $, $ x_{1}=\psi\left(x\right) $, $ y_{1}=\chi\left(y\right) $. Therefore
this function is in fact a mapping from a product of two 1-dimensional
manifolds to a third $ 1 $-dimensional manifold. Let us denote them by $ L_{1} $,
$ L_{2} $, $ L_{3} $. If we work in the local situation we have a marked point,
therefore the coordinate changes $ \varphi $, $ \psi $, and $ \chi $ send 0 to 0. The
correspondences
\begin{equation}
L_{1}\to L_{3}\colon x\mapsto F\left(x,0\right),\qquad L_{2}\to L_{3}\colon y\mapsto F\left(0,y\right)
\notag\end{equation}
determine identifications of $ L_{1} $ and $ L_{2} $ with
$ L_{3} $. So let $ L_{1}=L_{2}=L_{3}=L $. Consider a coordinate system on $ L $. Now basing on a
Veronese web
we have constructed a function $ \widetilde F\left(x,y\right) $ up to a change of the
form $ \widetilde F_{1}\left(x,y\right)=\phi\left(\widetilde F\left(\phi^{-1}\left(x\right),\phi^{-1}\left(y\right)\right)\right) $ with restrictions $ \widetilde F\left(x,0\right)=\widetilde F\left(0,x\right)=x $.
Consider a function $ G\left(x\right)=\widetilde F\left(x,x\right) $. It is defined up to a change
$ G_{1}\left(x\right)=\phi\left(\widetilde G\left(\phi^{-1}\left(x\right)\right)\right) $, and $ \frac{dG}{dx}|_{x=0}=2 $. We can find $ \phi $ such
that $ G_{1}\left(x\right)=2x $. Now {\em the only changes of\/} $ \phi $ {\em that preserve this restriction\/} are
linear changes. This gives us {\em a canonically defined
function\/} $ \overline F $ with conditions $ \overline F\left(x,0\right)=\overline F\left(0,x\right)=x $, $ \overline F\left(x,x\right)=2x $ up to a change
$ \overline F_{1}\left(x,y\right)=\alpha^{-1}\overline F\left(\alpha x,\alpha y\right) $. Therefore we can write
\begin{equation}
\overline F\left(x,y\right)=x+y+xy\left(x-y\right)\overset{\,\,{}_\circ} F\left(x,y\right),
\notag\end{equation}
where the function $ \overset{\,\,{}_\circ} F $ is arbitrary and defined up to a change
\begin{equation}
\overset{\,\,{}_\circ} F\left(x,y\right)\mapsto\overset{\,\,{}_\circ} F_{1}\left(x,y\right)=\alpha^{2}\overset{\,\,{}_\circ} F\left(\alpha x,\alpha y\right),\quad \alpha\not=0.
\notag\end{equation}
\end{remark}
Therefore we have proved the following
\begin{theorem}
\begin{enumerate}
\item
The set of germs of $ 2 $-dimensional Veronese webs (or
$ 3 $-dimensional bihamiltonian manifolds) up to
isomorphism can be identified with the set of germs of functions of
two variables up to a change
\begin{equation}
\overset{\,\,{}_\circ} F_{1}\left(x,y\right)=\alpha^{2}\overset{\,\,{}_\circ} F\left(\alpha x,\alpha y\right),\quad \alpha\not=0.
\notag\end{equation}
\item
The corresponding to $ \overset{\,\,{}_\circ} F $ web can be written in an appropriate
coordinate system as three foliations given by equations
\begin{equation}
x=\operatorname{const},\quad y=\operatorname{const},\quad x+y+xy\left(x-y\right)\overset{\,\,{}_\circ} F\left(x,y\right)=\operatorname{const}
\notag\end{equation}
correspondingly. This coordinate system is determined uniquely up to a
homotety if $ \overset{\,\,{}_\circ} F=0 $ and uniquely otherwise.
\item
The brackets of the corresponding to $ \overset{\,\,{}_\circ} F $ bihamiltonian manifold can be
written in an appropriate coordinate system as
\begin{equation}
\begin{aligned}
\left\{f\left(x,y,z\right),g\left(x,y,z\right)\right\}_{1} & =\left(1+xy\left(x-y\right)\overset{\,\,{}_\circ} F_{x}+y\left(2x-y\right)\overset{\,\,{}_\circ} F\right)\left(-f_{y}g_{z}+f_{z}g_{y}\right),
\\
\left\{f\left(x,y,z\right),g\left(x,y,z\right)\right\}_{2} & =\left(1+xy\left(x-y\right)\overset{\,\,{}_\circ} F_{y}+x\left(x-2y\right)\overset{\,\,{}_\circ} F\right)\left(f_{x}g_{z}-f_{z}g_{x}\right).
\end{aligned}
\notag\end{equation}
This coordinate system is determined uniquely up to a transformation
\begin{equation}
x_{1}=x,\quad y_{1}=y,\quad z_{1}=z+\beta\left(x,y\right)
\notag\end{equation}
if $ \overset{\,\,{}_\circ} F\not=0 $ and up to a transformation
\begin{equation}
x_{1}=\alpha x,\quad y_{1}=\alpha y,\quad z_{1}=\alpha^{-1}z+\beta\left(x,y\right)
\notag\end{equation}
otherwise.
\end{enumerate}
\end{theorem}
\section{Appendix on linear algebra }\label{h3}\myLabel{h3}\relax
Consider a pair of skewsymmetric bilinear forms $ \alpha $, $ \beta $ in a
(finite-dimensional) vector space $ V $. Call a pair of forms {\em decomposable\/} if
there exist two supplementary non-zero subspaces $ V_{1} $, $ V_{2} $ such that both
forms $ \alpha $ and $ \beta $ are direct sums of their restrictions on $ V_{1} $ and $ V_{2} $, i.e.,
the subspaces $ V_{1} $ and $ V_{2} $ are skeworthogonal\footnote{I.e., orthogonal with respect to a skew form.} with respect to both forms.
Any pair of forms in a vector space $ V $ can be decomposed in a direct
sum of {\em undecomposable\/} pairs in subspaces $ V_{i} $. We want to describe pairs
of forms in a vector space up to an isomorphism (i.e., a coordinate
change in the space $ V $). It is sufficient to describe undecomposable
pairs.
As it turns out, it is useful to describe such pairs basing on other
objects of linear algebra: pairs of linear mappings from one vector space
to another. It is clear what is a direct sum of two such pairs. Let us
call a pair {\em undecomposable\/} if it cannot be represented as a direct sum.
\begin{theorem}[\cite{GelZakh89Spe}] \label{th3.1}\myLabel{th3.1}\relax
\begin{enumerate}
\item
The list of undecomposable components
(up to an isomorphism) of a pair
of skewsymmetric bilinear forms is uniquely defined, the same is true for
a pair of linear mappings;
\item
If a pair of skewsymmetric bilinear forms $ \alpha $, $ \beta $
in a
(finite-dimensional) vector space $ V $ is undecomposable, then the
vector space $ V $ can be represented as a direct sum of two subspaces $ W_{1} $ and
$ W_{2} $, where
\begin{enumerate}
\item
Both $ W_{1} $ and $ W_{2} $ are isotropic with respect to both forms $ \alpha $ and $ \beta $;
\item
The pairings $ \alpha $ and $ \beta $ determine two mappings $ \widetilde\alpha,\widetilde\beta\colon W_{1}\to W_{2}^{*} $, and this
pair of mappings from one vector space to another is undecomposable in
the above sense;
\end{enumerate}
\item
On the other side, any undecomposable pair of mappings $ \widetilde\alpha,\widetilde\beta:
W_{1}\to W_{2}^{*} $ determines an undecomposable pair of skewsymmetric bilinear
forms $ \alpha $, $ \beta $ in the vector space $ W_{1}\oplus W_{2} $ by the rule
\begin{equation}
\begin{aligned}
\alpha\left(\left(w_{1},w_{2}\right),\left(w'_{1},w'_{2}\right)\right) & = \left< \widetilde\alpha\left(w_{1}\right), w'_{2} \right> - \left< \widetilde\alpha\left(w'_{1}\right), w_{2} \right>,
\\
\beta\left(\left(w_{1},w_{2}\right),\left(w'_{1},w'_{2}\right)\right) & = \left< \widetilde\beta\left(w_{1}\right), w'_{2} \right> - \left< \widetilde\beta\left(w'_{1}\right), w_{2} \right>.
\end{aligned}
\notag\end{equation}
\item
Any undecomposable pair of mappings from a vector space $ X_{1} $ to
a vector space $ X_{2} $ is isomorphic to exactly one pair from the list:
\begin{enumerate}
\item
The Jordan case $ J_{k}^{\lambda} $, $ k\geq1 $ with {\em eigenvalue\/} $ \lambda $: here $ X_{1}=X_{2} $, $ \dim X_{1}=k $,
$ \widetilde\alpha=\operatorname{id}_{X_{1}} $, $ \widetilde\beta $ is a
mapping from $ X_{1} $ to $ X_{1} $ with exactly one Jordan block (of size $ k $) with
eigenvalue $ \lambda $;
\item
The Jordan case $ J_{k}^{\infty} $, $ k\geq1 $ with {\em eigenvalue\/} $ \infty $: here $ X_{1}=X_{2} $, $ \dim X_{1}=k $, $ \widetilde\alpha $
is a mapping from $ X_{1} $ to $ X_{1} $ with exactly one Jordan block (of size $ k $)
with eigenvalue 0, $ \widetilde\beta=\operatorname{id}_{X_{1}} $;
\item
The Kroneker case $ K_{k}^{+} $, $ k\geq1 $: here $ X_{1}=S^{k-1}R $, $ X_{2}=S^{k}R $ (i.e., the
symmetrical powers), $ R $ is a $ 2 $-dimensional vector space with a basis $ r_{1} $,
$ r_{2} $, $ \widetilde\alpha = M_{r_{1}} $, $ \widetilde\beta = M_{r_{2}} $, where $ M_{r} $ is the mapping of multiplication by $ r $
from $ S^{k-1}R $ to $ S^{k}R $;
\item
The Kroneker case $ K_{k}^{-} $, $ k\geq1 $: here $ X_{1}=S^{k}R $, $ X_{2}=S^{k-1}R $ (i.e., the
symmetrical powers), $ R $ is a $ 2 $-dimensional vector space with a basis $ r_{1} $,
$ r_{2} $, $ \widetilde\alpha = D_{r_{1}} $, $ \widetilde\beta = D_{r_{2}} $, where
\begin{equation}
D_{r_{1}}=\frac{\partial}{\partial r_{1}},D_{r_{2}}=\frac{\partial}{\partial r_{2}}\colon S^{k}R \to S^{k-1}R;
\notag\end{equation}
\item
The trivial Kroneker case $ K_{0}^{+}\colon \dim X_{1}=0 $, $ \dim X_{2}=1 $, $ \widetilde\alpha=\widetilde\beta=0 $;
\item
The trivial Kroneker case $ K_{0}^{-}\colon \dim X_{1}=1 $, $ \dim X_{2}=0 $, $ \widetilde\alpha=\widetilde\beta=0 $;
\end{enumerate}
\item
If a pair of skewsymmetric bilinear forms is in general position,
then
\begin{enumerate}
\item
if the space $ V $ is even-dimensional all the undecomposable components
are $ 2 $-dimensional, canonically defined and correspond to the pairs of
mappings $ J_{1}^{\lambda} $, $ \lambda\in{\mathbb C}\cup\left\{\infty\right\} $;
\item
if the space $ V $ is odd-dimensional, $ \dim V=2k-1 $, then there is
only one undecomposable component (so the pair is undecomposable),
corresponding to the Kroneker case $ K_{k}^{-} $
(or $ K_{k}^{+} $, since $ K_{k}^{+} $ and $ K_{k}^{-} $ lead to isomorphic pairs of skew-symmetric
bilinear form);
\end{enumerate}
\item
If an undecomposable pair of skewsymmetric bilinear forms in
an odd-dimensional
vector space $ V $ corresponds (as above) to the Kroneker pair of mappings
$ K_{k}^{-}\colon W_{1}\to W_{2}^{*} $, then the subspace $ W_{1}\subset V $ is canonically defined. It is
spanned by $ 1 $-dimensional kernels (i.e., by the vectors which are orthogonal
to the whole space) of linear combinations $ \alpha-\lambda\beta $ of forms $ \alpha $ and $ \beta $. These
kernels considered as points in the projectivization $ PW_{1} $ of the space $ W_{1} $
form {\em a Veronese curve}, i.e., a curve of minimal possible degree (equal to
$ \dim PW_{1} $) spanning the whole space $ PW_{1} $.
\end{enumerate}
\end{theorem}
We see that there is a close relation between pairs of linear
mappings and pairs of skewsymmetric bilinear forms. If we consider
a pair of bilinear forms in a vector space $ V $ as a pair of mappings $ V\to V^{*} $,
then this pair of mappings becomes a sum of two (dual)
pairs of
mappings, and the original pair of forms can be reconstructed (up to an
isomorphism) basing on any one of these dual pairs.\footnote{It is the place to note that the analogue of this theorem for {\em symmetric\/}
bilinear forms is wrong, as shows an example of a pair of forms on
$ 1 $-{\em dimensional\/} vector space. In fact in the symmetric case there is one
additional series of
undecomposable pairs that includes this example. All other undecomposable
pairs can be constructed basing on pairs of linear mappings.} If the pair of forms
is undecomposable, the corresponding pairs of mappings is undecomposable
too, and visa versa. We can call
eigenvalues and sizes of Jordan blocks in the corresponding pair of mappings
{\em eigenvalues and sizes of blocks\/} for a pair of skewsymmetric forms.
\section{Appendix on flats and maps }\label{h4}\myLabel{h4}\relax
We want to prove here that if we consider two projections $ \pi_{1} $, $ \pi_{2} $
from the incidence set $ C\subset M\times M^{\left(2\right)} $ on the factors in the case of
an even-dimensional $ M $, then for any regular point $ m\in M $
the ideal $ \pi_{2*}\pi_{1}^{*}I_{m} $ on $ M^{\left(2\right)} $ is of codimension $ n=\frac{1}{2}\dim M $. Here $ I_{m} $ is the
corresponding to $ m\in M $ ideal in $ {\mathcal O}\left(M\right) $, $ M^{\left(2\right)} $ is the set of weak leaves in $ M $.
We have
already shown that this is true on an open dense subset of good points.
First of all we want to prove that the mapping $ \pi_{1} $ {\em is flat\/} on
the set of regular points. That means (in this case) that the codimension
of $ \pi_{1}^{*}I_{m} $ does not depend on the point $ m $, hence is indeed $ n $. We will use
the following properties of flat maps:
\begin{enumerate}
\item
An isomorphism is flat;
\item
If a map $ Y\to X $ is flat {\em locally\/} on $ X $, then it is flat;
\item
If the space of functions on $ Y $ is a finitely generated free module
over the ring of functions on $ X $, then the mapping $ Y\to X $ is flat;
\item
A composition of flat mappings is flat;
\item
If a mapping $ \alpha\colon Y\to X $ is flat and there is a mapping $ i $ from $ X' $ to $ X $,
then the {\em inverse image\/} of $ \alpha $
\begin{equation}
X'\times_{X}Y\xrightarrow[]{i^{*}\alpha} X'
\notag\end{equation}
is also flat.
\end{enumerate}
Let us consider instead of $ \pi_{1} $ some closely related mapping $ p_{1}:
C'\to M $. To define it we begin with a definition of $ C' $. Denote by $ \operatorname{Gr}_{2}T_{m}M $
the space of $ 2 $-dimensional subspaces in the $ T_{m}M $, let $ \operatorname{Gr}_{2}TM = \bigcup_{m\in M}
\operatorname{Gr}_{2}T_{m}M $. Let $ C'\subset\operatorname{Gr}_{2}TM $ be the subset consisting of tangent spaces to weak
leaves and $ p_{1} $ be a natural projection.
There is a natural map from the space $ C $ to $ C' $ that sends a pair
$ \left(m,L\right) $, $ m\in L $, to the subspace $ T_{m}L\subset T_{m}M $. It is easy to see that this mapping is
an isomorphism, so it is sufficient to show that $ p_{1} $ is flat. However, the
structure of the map $ p_{1} $ is much simpler, {\em since this mapping is an
inverse image from the space of pairs of skewsymmetric bilinear forms}.
Indeed, consider a local coordinate system on $ M $. It identifies
$ M $ with a piece of a vector space, call it $ V^{*} $. Now all the
cotangent spaces at different points of $ M $ are identified with $ V $.
Then to any point $ m\in M $ corresponds a regular pair of
skewsymmetric
forms in the vector space $ V $. Let $ U\subset\Lambda^{2}V^{*}\times\Lambda^{2}V^{*} $ be a subset of regular
pairs. Consider a subset $ {\mathcal C} $ of $ U\times\operatorname{Gr}_{2}V $ consisting of triples $ \left(\alpha,\beta,S\right) $ such
that $ S $ is a kernel of some non-zero linear combination $ \lambda\alpha+\mu\beta $. It is easy
to see
now that the mapping $ C'\to M $ is an inverse image of the mapping $ {\mathcal C}\to U $ with
respect to the map $ M\to U $.
So what remains to prove is the flatness of the mapping $ {\mathcal C}\to U $. Now we
want to consider yet another mapping $ {\mathcal D}\to U $. Here $ {\mathcal D} $ is a subset $ U\times{\mathbb C} $
consisting of triples $ \left(\alpha,\beta,\lambda\right) $ such that $ \lambda $ is an eigenvalue of the pair $ \alpha $,
$ \beta $, i.e., such that the form $ \alpha-\lambda\beta $ is degenerate. Since the pair $ \left(\alpha,\beta\right) $ is
regular, the kernel of the form $ \alpha-\lambda\beta $ is $ 2 $-dimensional, therefore there is
a natural isomorphism $ {\mathcal D}\to{\mathcal C} $. Now it is sufficient to prove that $ {\mathcal D}\to U $ is
flat.
The basic example of a flat mapping is the mapping $ A_{n}\to B_{n} $, where $ B_{n} $ is a
set of all polynomials $ P $ of degree $ n $ with the leading coefficient 1,
and $ A_{n} $
is a set of solutions, $ A_{n}=\left\{\left(x,P\right)\in{\mathbb C}\times B_{n} \mid P\left(x\right)=0\right\} $. The flatness of this
mapping is equivalent to a fact that any polynomial of degree $ n $ has
exactly $ n $ solutions, if counted with multiplicity. Using the above facts,
we can prove the flatness of this map by the note that any function $ f $ on
$ A_{n} $ can be uniquely represented in the form
\begin{equation}
f\left(x,P\right)= f_{0}\left(P\right) + xf_{1}\left(P\right)+\dots + x^{n-1}f_{n-1}\left(P\right),
\notag\end{equation}
therefore the space of functions on $ A_{n} $ is
indeed a free module over the ring of functions on $ B_{n} $.
Now we can apply this example to the proof of flatness of the
mapping $ {\mathcal D}\to U $. Let us consider the characteristic polynomial $ P_{\alpha,\beta}=\det \left(\alpha-\lambda\beta\right) $ of
the pair $ \left(\alpha,\beta\right) $. The theorem on linear algebra shows that this polynomial
can be represented as a square of a polynomial $ Q_{\alpha,\beta} $. We can normalize $ Q $ to
get a polynomial with the leading coefficient 1. In this way we have
defined a map $ U\to B_{n} $. It is clear that the map $ {\mathcal D}\to U $ is an inverse image of
the map $ A_{n}\to B_{n} $, so it is flat. Therefore, the map $ C\to M $ is indeed flat.
We proved that the codimension of
$ \pi_{1}^{*}I_{m} $ is $ n $.
What remains to prove is that the codimension of $ \pi_{2*}\pi_{1}^{*}I_{m} $ is equal
to the codimension of $ \pi_{1}^{*}I_{m} $. Speaking nonformally, this a consequence of
the fact that the ideal $ \pi_{1}^{*}I_{m} $ ``lives on the submanifold $ m\times M^{\left(2\right)}\subset M\times M^{\left(2\right)} $'',
and this submanifold projects isomorphically on $ M^{\left(2\right)} $. To give a
formal proof let us consider the inclusion
$ C\hookrightarrow M\times M^{\left(2\right)} $ and the ideal $ I_{C} $ consisting of vanishing on $ C $ functions. Let us
consider
this picture locally. The ring $ {\mathcal O}\left(C\right) $ is the quotient of $ {\mathcal O}\left(M\times M^{\left(2\right)}\right) $ by $ I_{C} $. Let
$ P_{1} $ and $ P_{2} $ be two projections from $ M\times M^{\left(2\right)} $ to the factors. Then $ \pi_{1}^{*}I_{m} \subset{\mathcal O}\left(C\right) $
is just $ P_{1}^{*}I_{m}/ \left(I_{C}\cap P_{1}^{*}I_{m}\right) $, hence $ {\mathcal O}\left(C\right)/\pi_{1}^{*}I_{m} $ coincides
with $ {\mathcal O}\left(M\times M^{\left(2\right)}\right)/\left(I_{C}+P_{1}^{*}I_{m}\right) $.
However, the last ring can be written as
\begin{equation}
\left({\mathcal O}\left(M\times M^{\left(2\right)}\right)/P_{1}^{*}I_{m}\right)/\left(\left(I_{C}+P_{1}^{*}I_{m}\right)/P_{1}^{*}I_{m}\right)={\mathcal O}\left(m\times M^{\left(2\right)}\right)/\left(I_{C}+P_{1}^{*}I_{m}\right).
\notag\end{equation}
A function on $ M^{\left(2\right)} $ is in the ideal $ \pi_{2*}\pi_{1}^{*}I_{m} $ if its inverse image on
$ M\times M^{\left(2\right)} $ is in the ideal $ \pi_{1}^{*}I_{m} $, i.e., the image of this function in the
ring $ {\mathcal O}\left(m\times M^{\left(2\right)}\right)/\left(I_{C}+P_{1}^{*}I_{m}\right) $ is 0. Since the ring
$ {\mathcal O}\left(m\times M^{\left(2\right)}\right) $ is isomorphic to the ring $ {\mathcal O}\left(M^{\left(2\right)}\right) $, the ideal $ \pi_{2*}\pi_{1}^{*}I_{m} $
corresponds under this isomorphism to the ideal $ \left(I_{C}+P_{1}^{*}I_{m}\right)/P_{1}^{*}I_{m} $,
therefore has the same codimension. This completes the proof.
\section{Appendix on global bihamiltonian geometry }\label{h5}\myLabel{h5}\relax
We have seen in the section on the Kodaira theorem that to
construct a
local odd-dimensional bihamiltonian structure we should just have a
nonlinear bundle of rank 1 over a projective line.
However, to do the same in the
even-dimensional case we need several $ 2 $-dimensional bihamiltonian
systems. Therefore in the odd-dimensional case we should
have only the information of ``topological'' origin (i.e., a
complex structure on a given topological object), and not of the
differential-geometrical origin, as in the even-dimensional case.
However, in the global case the picture can be very similar to that
``topological paradise'' even in even-dimensional case, as shows the following
\begin{lemma} Let $ M $ be a manifold such that
\begin{equation}
\dim \Gamma\left(\Lambda^{2}TM\right)=k<\infty\text{, }\dim \Gamma\left(\Lambda^{3}TM\right)=0.
\notag\end{equation}
Then on $ M $ there is a canonical (up to a linear change) $ k $-hamiltonian
structure. \end{lemma}
\begin{proof} Let us remind that a Poisson structure is a bracket on the set
of functions that satisfies the Leibniz and Jacobi condition. As we have
seen, any bracket satisfying the Leibniz condition can be written as
\begin{equation}
\left\{f,g\right\}|_{x}=\left< \eta|_{x}, \left(df\wedge dg\right)|_{x} \right>
\notag\end{equation}
for an appropriate bivector field $ \eta $ on $ M $ (i.e., a section of $ \Lambda^{2}TM $). Let
us consider the Jacobi condition for this bracket. It is easy to see that
the number
\begin{equation}
{\mathcal H}\left(f,g,h\right)|_{x} = \left(\left\{\left\{f,g\right\},h\right\}+\left\{\left\{g,h\right\},f\right\}+\left\{\left\{h,f\right\},g\right\}\right)|_{x}
\notag\end{equation}
depends only on the differentials of functions $ f $, $ g $, $ h $ in the point $ x $,
and is skewsymmetric with respect to these differentials, so it can be
written as
\begin{equation}
{\mathcal H}\left(f,g,h\right)|_{x} = \left< {\mathcal H}|_{x}, \left(df\wedge dg\wedge dh\right)|_{x} \right>
\notag\end{equation}
for some $ 3 $-vector field $ {\mathcal H} $ on $ M $ (i.e., a section of $ \Lambda^{3}M $). Therefore the
condition of the lemma implies that any global bivector field on $ M $ gives rise
to a Poisson structure.
That means that we have $ k $-dimensional vector space of Poisson
brackets on $ M $, i.e., a $ k $-hamiltonian structure.\end{proof}
\begin{remark} In fact this $ k $-hamiltonian structure is defined canonically up
to a linear change of the $ k $ basic Poisson structures, but this is {\em the
object\/} people usually work with. \end{remark}
\begin{example} Let us consider a $ 2 $-dimensional case. Then $ \dim \Gamma\left(\Lambda^{3}M\right) $ is 0 for
sure, so we should bother only with $ \Gamma\left(\Lambda^{2}M\right) $. Let us consider the behavior
of the bivector field $ \frac{d}{dx}\wedge\frac{d}{dy} $ on the infinity in $ {\mathbb P}^{2} $. The
corresponding $ 2 $-form $ dx\wedge dy $ has a pole of the third order on infinity: in
the coordinates $ \left(x:y:1\right)=\left(1:z:t\right) $ it can be written as
$ d\frac{1}{t}\wedge d\frac{z}{t}=-\frac{1}{t^{3}}dt\wedge dz. $ Therefore the bivector field has a zero of
the third order on infinity (since in local frames the bivector field and
the $ 2 $-form have mutually inverse coefficients).
This means that the global sections of $ \Lambda^{2}T{\mathbb P}^{2} $ can be written as $ P_{3}\left(x,y\right)
\frac{d}{dx}\wedge\frac{d}{dy} $, where $ P_{3} $ is a cubic polynomial. Therefore {\em on the
projective plane a natural\/} $ 10 $-{\em hamiltonian structure is defined}. \end{example}
\begin{example} Since we are primary interested in {\em bihamiltonian\/} structures,
we give here another example. Consider two different cubic curves on $ {\mathbb P}^{2} $.
They intersect one another in 9 points of the plane. Consider a blow-up
$ M $ of the plane in these 9 points. A bivector field $ \eta $ on $ M $ corresponds to a
bivector field $ \widetilde\eta $ on $ {\mathbb P}^{2} $ at least outside of these points. However, the
Hartogs theorem implies that the bivector field $ \widetilde\eta $ can be extended to
the whole $ {\mathbb P}^{2} $.
We have shown already that a nondegenerate polyvector on a
$ 2 $-dimensional
manifold gets a pole when raised to a blow-up. This implies that
the bivector field $ \widetilde\eta $ on $ {\mathbb P}^{2} $ has zeros in these 9 points on the plane.
Therefore the
corresponding to
$ \widetilde\eta $
cubic polynomial on the plane has zeros in these points,
therefore is a linear combination of equations of initial cubic curves.
That means that {\em a canonical\/} $ 2 $-{\em bihamiltonian structure is
defined on\/} $ M $. \end{example}
\begin{remark} It is easy to see that this bihamiltonian structure is {\em in
general position\/} in a neighborhood of any point on $ M $. However, the space of
cubic polynomials vanishing in given 8 points (in general position) on the
plane is also $ 2 $-dimensional. That means that instead of blowing-up 9
points it were sufficient to blow-up only 8 points of these $ 9 $---on the
resulting manifold there is a natural bihamiltonian structure.
There is a remarkable algebro-geometrical construction on a plane
that to a $ 8 $-tuple of points on $ {\mathbb P}^{2} $ in general position associates a $ 9 $th
point: any cubic passing through these 8 points passes
through
this $ 9 $th point.
(Therefore we cannot get an arbitrary $ 9 $-tuple of points as an
intersection of two cubics!)
Now consider the result $ M' $ of blowing up the plane at these 8
points. This manifold has two independent global bivector fields, and they
both vanish at the $ 9 $th point.
Therefore the bihamiltonian structure is not in general position in
a neighborhood of this point. \end{remark}
Now, when we have constructed a $ 2 $-dimensional manifold $ M $ with a
canonically defined bihamiltonian structure, we can consider the Hilbert
scheme $ S^{n}M $ (the definition of the Hilbert scheme can be found in the
section~\ref{h1}). As it was shown in the section~\ref{s1.4}, the bihamiltonian
structure on $ M $ determines a bihamiltonian
structure on $ S^{n}M $. We can show that if we are sufficiently lucky this
bihamiltonian structure on $ S^{n}M $ is also canonically defined.
\begin{lemma} Consider a connected $ 2 $-dimensional manifold $ M $ such that
$ \dim \Gamma\left(TM\right)=0 $, $ \dim \Gamma\left({\mathcal O}\left(M\right)\right)=1 $. Then any bivector field
on the Hilbert scheme $ S^{n}M $ corresponds (in
the specified above way) to a bivector field on $ M $. Moreover,
$ \dim \Gamma\left(\Lambda^{3}TS^{n}M\right)=0 $. \end{lemma}
\begin{proof} Fix $ n-1 $ different point $ m_{1},m_{2},\dots ,m_{n-1} $ on $ M $ and consider
another (variable) point $ m_{0}\in M $. A neighborhood of the point
$ \left\{m_{0},m_{1},m_{2},\dots ,m_{n-1}\right\} $ on $ S^{n}M $ can be considered as a direct product of
neighborhoods of points $ m_{0} $, $ m_{1} $, $ m_{2},\dots . $ This decomposition associates to
any bivector at $ \left\{m_{0},m_{1},m_{2},\dots ,m_{n-1}\right\}\in S^{2}M $ a set of bivectors, one at any
point $ m_{0} $, $ m_{1} $,
$ m_{2},\dots $, $ m_{n-1} $, and a set of elements of tensor products $ T_{m_{i}}M\otimes T_{m_{j}}M $. Consider
a global bivector field on $ S^{n}M $ and the first component of the value of
this bivector field at $ \left\{m_{0},m_{1},m_{2},\dots ,m_{n-1}\right\} $, that is a bivector in the
point $ m_{0}\in M $. We defined a bivector field on $ M\smallsetminus\left\{m_{1},m_{2},\dots ,m_{n-1}\right\} $. However,
by the Hartogs theorem again, this field can be extended to the whole $ M $.
This bivector field is a linear combination of basic bivector fields
on $ M $. The coefficients of this combination depend on $ m_{1},m_{2},\dots ,m_{n-1} $.
However, if we consider a coefficient as a function of, say, $ m_{1} $, we see
that it is defined anywhere outside of $ m_{2},\dots ,m_{n-1} $, therefore it can be
extended to a global function and is constant. That means that we defined
a global bivector field on $ M $ basing on a global bivector field on $ S^{n}M $. It
is easy to see now that this is an inverse map to the construction of
bivector field on $ S^{n}M $ basing on a bivector field on $ M $.
Moreover, let us fix a cotangent to $ M $ vector at $ m_{i} $ for a fixed $ i $.
Consider the component in $ T_{m_{i}}M\otimes T_{m_{0}}M $ of the value of the bivector field
and the ``scalar product'' of this tensor with the fixed covector at $ m_{i} $. In
this way we get a tangent vector at $ m_{0} $. This determines a vector field on
$ M\smallsetminus\left\{m_{1},m_{2},\dots ,m_{n-1}\right\} $, that again can be extended to the whole $ M $. Therefore
the corresponding vector field is 0. Hence the off-diagonal components
of the bivector field vanish, so it is determined by the diagonal
components.
The same argument shows that any $ 3 $-vector field on $ S^{n}M $ should be 0. \end{proof}
In the above $ 2 $-dimensional examples of bihamiltonian manifolds there
is no global vector fields and global functions, therefore the
corresponding Hilbert schemes are also equipped with canonically defined
bihamiltonian structures. So:
\begin{theorem} Fix 8 point on a plane in general position (here that means that
they are not on the same conic and no 5 point subsets is on the same
line). Consider a $ 9 $-th point that is the only other point of intersection
of two cubics passing through these 8 points. Denote the blow-up of the
plane in these 8 points by $ M_{1} $, in all 9 points by $ M_{2} $. Then on the Hilbert
schemes $ S^{n}M_{1} $, $ S^{n}M_{2} $ the spaces of global bivector fields are
$ 2 $-dimensional, and any bivector field determines a Poisson structure.
Therefore on both these $ 2n $-dimensional manifolds a canonical
bihamiltonian structure is defined, the set of (generalized\footnote{See section~\ref{s1.7}.}) weak leaves is
isomorphic
to the corresponding $ 2 $-dimensional manifold $ M_{1} $ or $ M_{2} $ and these
bihamiltonian structures can be reconstructed basing on the canonically
defined bihamiltonian structures on $ M_{1} $, $ M_{2} $. \end{theorem}
\begin{remark} As we will see in the section~\ref{s6.4}, a neighborhood of any point on
these examples of bihamiltonian manifolds is also an example of a
local bihamiltonian manifold to which we can apply a weak classification
theorem from the section~\ref{s1.7}. Therefore we found examples of global
bihamiltonian manifolds that are classifiable in a neighborhood of any
point. Of course, in these cases the classification theorem shows only
that we can reconstruct $ M $ basing on $ S^{n}M $. \end{remark}
\section{Appendix on the local geometry of a bihamiltonian Hilbert scheme }\label{h6}\myLabel{h6}\relax
\subsection{Preliminaries }\label{s6.1}\myLabel{s6.1}\relax We have shown above that a bihamiltonian
structure in a neighborhood
of a regular point\footnote{Let us remind that the regular point is a point where two tensors of
Poisson structures form a regular pair, i.e., the dimension of the
stabilizer of this pair in $ \operatorname{GL}\left(T_{m}M\right) $, $ m\in M $ is the minimal possible.} is isomorphic to a
bihamiltonian structure on a Hilbert scheme
(under a mild assumption). However, the given point
goes under this correspondence to a regular point of the Hilbert scheme.
It is easy to see (even in $ 4 $-dimensional example above) that not any
point of the Hilbert scheme is a regular point.
\begin{example} Consider the coordinates $ X $, $ Y $ on the plane $ M $ with Poisson
structures $ \frac{d}{dX}\wedge\frac{d}{dY} $ and $ X\frac{d}{dX}\wedge\frac{d}{dY} $ and corresponding coordinates
\begin{equation}
\xi=\frac{1}{\sqrt{2}}\left(X_{1}+X_{2}\right)\text{, }\eta=\frac{1}{\sqrt{2}}\left(Y_{1}+Y_{2}\right)\text{, }x=\frac{1}{\sqrt{2}}\left(X_{1}-X_{2}\right)\text{, }y=\frac{1}{\sqrt{2}}\left(Y_{1}-Y_{2}\right)
\notag\end{equation}
on the $ M\times M $. We know that $ S^{2}M $ is a blow-up of $ M\times M/{\mathfrak S}_{2} $ in the diagonal.
Consider a point on $ S^{2}M $ on the intersection of the preimage of the
diagonal and of the preimage of $ x=0 $. Then $ \xi $, $ \eta $, $ \widetilde\alpha=x/y $ and $ \widetilde\beta=y^{2} $
form a coordinate frame in this point. (In the example in the
section~\ref{s7} we
considered coordinates $ \alpha=y/x $, $ \beta=x^{2} $ in the remaining points of the
exceptional divisor.)
The first Poisson structure on $ M\times M $ corresponds to the bivector field
\begin{equation}
\frac{\partial}{\partial\xi}\wedge\frac{\partial}{\partial\eta}+\frac{\partial}{\partial x}\wedge\frac{\partial}{\partial y},
\notag\end{equation}
hence the first Poisson structure on $ S^{2}M $ corresponds to the bivector
field
\begin{equation}
\frac{\partial}{\partial\xi}\wedge\frac{\partial}{\partial\eta}+2\frac{\partial}{\partial\widetilde\alpha}\wedge\frac{\partial}{\partial\widetilde\beta}.
\notag\end{equation}
To find the expression of the second structure it is more suitable to work
with the corresponding symplectic structure $ \frac{1}{X}dX\wedge dY. $ We need to
express $ \frac{1}{X_{1}}dX_{1}\wedge dY_{1} + \frac{1}{X_{2}}dX_{2}\wedge dY_{2} $ in the frame $ \xi $, $ \eta $, $ \widetilde\alpha $, $ \widetilde\beta $.
A tedious computation shows that this form coincides with
\begin{equation}
\frac{\sqrt{2}}{\xi^{2}-\widetilde\alpha^{2}\widetilde\beta}\left(\xi\,d\xi\,d\eta-\widetilde\alpha\widetilde\beta\,d\widetilde\alpha\,d\eta-\frac{\widetilde\alpha^{2}}{2}\,d\widetilde\beta\,d\eta-\frac{\widetilde\alpha}{2}\,d\xi\,d\widetilde\beta+\frac{\xi}{2}\,d\widetilde\alpha\,d\widetilde
\beta\right).
\notag\end{equation}
The corresponding Poisson structure can be calculated by inversion of
this ``matrix'' and is equal to
\begin{equation}
\frac{1}{\sqrt{2}}\left(\xi\frac{\partial}{\partial\xi}\wedge\frac{\partial}{\partial\eta}-\widetilde\alpha^{2}\frac{\partial}{\partial\xi}\wedge\frac{\partial}{\partial\alpha}+2\widetilde\alpha\widetilde\beta\frac{\partial}{\partial\xi}\wedge\frac{\partial}{\partial\widetilde\beta}-\widetilde\alpha\frac{\partial}{\partial\eta}\wedge\frac{\partial}{\partial\widetilde
\alpha}+2\xi\frac{\partial}{\partial\widetilde\alpha}\wedge\frac{\partial}{\partial\widetilde\beta}\right).
\notag\end{equation}
Therefore the corresponding recursion operator has in the basis $ d\xi $, $ d\eta $,
$ d\widetilde\alpha $, $ d\widetilde\beta $ the matrix
\begin{equation}
\frac{1}{\sqrt{2}}\left(
\begin{matrix}
\xi & 0 & \widetilde\alpha & 0
\\
0 & \xi & -\widetilde\alpha^{2} & 2\widetilde\alpha\widetilde\beta
\\
\widetilde\alpha\widetilde\beta & 0 & \xi & 0
\\
\widetilde\alpha^{2}/2 & \widetilde\alpha/2 & 0 & \xi
\end{matrix}
\right).
\notag\end{equation}
The characteristic polynomial is $ \left(\left(\xi-\lambda\right)^{2}-\widetilde\alpha^{2}\widetilde\beta\right)^{2} $. When $ \widetilde\alpha^{2}\widetilde\beta\not=0 $, the
corresponding pair of forms is decomposable, since the multiplicity of
eigenvalues of the recursion operator is only 2. However, if $ \widetilde\alpha=0 $, then
the recursion operator is diagonal, therefore the corresponding pair of
forms are proportional, therefore this pair is not regular. Therefore the
set of regular points on $ S^{2}M $ coincides with the set $ \widetilde\alpha\not=0 $. \end{example}
So the reasonable question is to describe all the regular points on
the Hilbert scheme. We can note first that it is sufficient to consider
points of the Hilbert scheme (i.e., ideals on the $ 2 $-dimensional manifold)
that do not come from products of previous Hilbert schemes (i.e.,
to consider ideals contained in exactly one maximal
ideal). So let $ A $ be a ring of functions on a neighborhood $ U $ of the given
point $ m\in M $ and $ I $ be an ideal such that $ A/I $ is a local ring with support at
$ m $.
Let us compute the tangent space to the Hilbert scheme $ S^{n}U $ of $ U $ at
the point $ I\in S^{n}U $. Let us remind that the Hilbert space is sitting inside
the Grassmannian $ \operatorname{Gr}_{n}\left(A\right) $ of subspaces of codimension $ n $ in $ A $ and that it
consists
of subspaces that are ideals in $ A $. Hence the tangent space to (the
smooth submanifold) $ S^{n}U $ is sitting inside the tangent space to this
Grassmannian. So we get the inclusion
\begin{equation}
T_{I}S^{n}U\hookrightarrow T_{I}\operatorname{Gr}_{n}\left(A\right)= \operatorname{Hom}_{{\mathbb C}}\left(I,A/I\right).
\notag\end{equation}
A standard theorem on the geometry of a Hilbert scheme (see section~\ref{s7}) shows
that the image of
this inclusion coincides with the set of $ A $-homomorphisms, i.e., with
\begin{equation}
\operatorname{Hom}_{A}\left(I,A/I\right)=\operatorname{Hom}_{A}\left(I/I^{2},A/I\right)=\operatorname{Hom}_{A/I}\left(I/I^{2},A/I\right).
\notag\end{equation}
Now the analysis of the section on Hilbert schemes shows that to any
Poisson structure $ \left\{,\right\} $ on $ M $ there corresponds a Poisson structure on the
Hilbert scheme, and if the initial Poisson structure is nondegenerate,
the Poisson structure on the Hilbert scheme is also nondegenerate.
A value at $ I\in S^{n}M $ of the bivector field
on $ S^{n}M $ that corresponds to this Poisson structure is an element of
$ \Lambda^{2}\operatorname{Hom}_{A/I}\left(I/I^{2},A/I\right) $. However, it is very difficult to write this element
explicitly.
We will use the fact that any Poisson structure $ \left\{,\right\} $ on $ 2 $-dimensional
manifold $ M $ with local coordinates $ x $ and $ y $ is proportional to a {\em standard\/}
Poisson structure $ \left\{,\right\}_{0} $ on it:
\begin{equation}
\left\{f,g\right\}|_{x}=\varphi\left(x\right)\left( \frac{\partial f}{\partial x}\frac{\partial g}{\partial y} - \frac{\partial g}{\partial x}\frac{\partial f}{\partial y} \right)\buildrel{\text{def}}\over{=}\varphi\left(x\right)\left\{f,g\right\}_{0}.
\notag\end{equation}
Let us denote the corresponding to $ \left\{,\right\}_{0} $ bivector in $ \Lambda^{2}\operatorname{Hom}_{A/I}\left(I/I^{2},A/I\right) $ by
$ \Phi_{0} $, the corresponding to $ \left\{,\right\} $ bivector by $ \Phi $. What we are going to do is to
express $ \Phi $ basing on $ \Phi_{0} $ and $ \varphi $.
We note first that the vector space $ \operatorname{Hom}_{A/I}\left(I/I^{2},A/I\right) $ is in fact an
$ A/I $-module. We want to consider the space $ \Lambda^{2}\operatorname{Hom}_{A/I}\left(I/I^{2},A/I\right) $ as a
subspace of
\begin{equation}
\operatorname{Hom}_{A/I}\left(I/I^{2},A/I\right)\otimes\operatorname{Hom}_{A/I}\left(I/I^{2},A/I\right).
\notag\end{equation}
We can define a structure
of an $ A/I $-module on the latter space, where $ a\cdot\left(\alpha\otimes\beta\right)=\left(a\cdot\alpha\right)\otimes\beta $. That action
does not preserve the subspace $ \Lambda^{2}\operatorname{Hom}_{A/I}\left(I/I^{2},A/I\right) $, however,
\begin{lemma} Consider the element $ \Phi_{0}\in\operatorname{Hom}_{A/I}\left(I/I^{2},A/I\right)\otimes\operatorname{Hom}_{A/I}\left(I/I^{2},A/I\right) $.
\begin{enumerate}
\item
For any element $ a\in A/I $ the tensor $ a\Phi_{0} $ is skewsymmetric;
\item
The bivector $ \Phi $ considered as an element of
\begin{equation}
\operatorname{Hom}_{A/I}\left(I/I^{2},A/I\right)\otimes\operatorname{Hom}_{A/I}\left(I/I^{2},A/I\right)
\notag\end{equation}
is equal to $ \phi\Phi_{0} $.
\end{enumerate}
\end{lemma}
\begin{proof} Let us consider first the case when the ideal $ I $ of codimension
$ n $ is a product of $ n $ different maximal ideals $ m_{i} $. In this case the lemma is
trivial, since we
can identify $ I/I^{2} $ with the direct sum of $ 2 $-dimensional
vector spaces $ \oplus_{i}m_{i}/m_{i}^{2} $ and
can identify $ A/I $ with the direct sum of $ 1 $-dimensional
algebras $ \oplus_{i}A/m_{i}\simeq\oplus_{i}{\mathbb C} $ acting in the first direct sum diagonally. Since the
bivectors $ \Phi $ and $ \Phi_{0} $ are diagonal with respect to this decomposition and
the diagonal blocks are proportional with coefficient $ \phi\left(m_{i}\right) $, the lemma is
true in this particular case.
In the general case we can represent any ideal as a limit of a family of
ideals of the considered above type, hence the lemma remains true. \end{proof}
\subsection{The description of the recursion operator }\label{s6.2}\myLabel{s6.2}\relax Let us remind the
definition of the {\em recursion operator}. We have two
bivectors $ \eta_{1} $, $ \eta_{2} $ in a point $ m $ of manifold $ M $. We can consider them as
elements of $ T_{m}M\otimes T_{m}M=\operatorname{Hom}\left(T_{m}^{*}M,T_{m}M\right) $. The recursion operator is defined in
the case when the bivector $ \eta_{1} $ corresponds to an invertible operator. We
define it as $ r=\eta_{2}\eta_{1}^{-1}\in\operatorname{End}\left(T_{m}M\right) $. In the general case, when the bivector $ \eta $
can be non-invertible, we can consider the
defined by this formula object as a {\em relation\/} in the space $ T_{m}M $, i.e., a
subspace in $ T_{m}M\times T_{m}M $ (when $ r $ is an operator, this subspace is the graph of
this operator).
\begin{corollary} Consider two Poisson structures $ \left\{,\right\}_{1} $, $ \left\{,\right\}_{2} $ on $ 2 $-dimensional
manifold $ M $ given by following formulae:
\begin{equation}
\left\{,\right\}_{i}=\varphi_{i}\left\{,\right\}_{0},\quad i=1,2.
\notag\end{equation}
(Here $ \left\{,\right\}_{0} $ is a nondegenerate Poisson structure on $ M $, $ \varphi_{1} $ and $ \varphi_{2} $ are two
functions on M.$ $) Then on the Hilbert
scheme $ S^{n}M $ we can consider two corresponding Poisson structures. The
corresponding recursion operator (or relation) in the tangent space to
the Hilbert scheme $ S^{n}M $ at $ I\in S^{n}M $ coincides with the operator (or relation)
\begin{equation}
\mu_{\varphi_{2}}\mu_{\varphi_{1}}^{-1}\in\operatorname{End}\left(\operatorname{Hom}_{A/I}\left(I/I^{2},A/I\right)\right),
\notag\end{equation}
where $ \mu_{\varphi} $ is the operator of multiplication by $ \varphi\in A $ in the $ A/I $-module
$ \operatorname{Hom}_{A/I}\left(I/I^{2},A/I\right) $. \end{corollary}
\subsection{The set of regular points on $ S^{n}M $ }\label{s6.3}\myLabel{s6.3}\relax Now we have sufficient
information to describe the
regular points of the bihamiltonian structure on the Hilbert scheme.
\begin{theorem} Let the point $ I\in S^{n}M $ is regular with respect to the
bihamiltonian structure on $ S^{n}M $ corresponding to the pair of Poisson
structures on $ M $.
\begin{enumerate}
\item
If the support of the ideal $ I $ on $ M $ consists of several points,
then the ideal $ I $ is a product of $ k $ ideals
$ I_{1},\dots ,I_{k} $ with supports at $ k $ different points of $ M $, any ideal $ I_{l} $,
$ l=1,\dots ,k $, is regular on the corresponding Hilbert scheme, and the
values of the ratio of Poisson structures on $ M $ at these points are
different;
\item
If the support of the ideal $ I $ on $ M $ is one point $ m\in M $, then there is a
smooth curve $ C\ni m $ on $ M $ such that $ I $ is a
direct image of an ideal on $ C $, i.e.,
\begin{equation}
f\in I \iff \left(\frac{d^{l}}{dt^{l}}f|_{C}\right)|_{m}=0\text{, }l=0,\dots ,\operatorname{codim} I-1.
\notag\end{equation}
Here $ t $ is a coordinate on $ C $. Moreover, if we write the second Poisson
structure on $ M $ as a multiple of the first one:
\begin{equation}
\left\{,\right\}_{2}=\varphi\left\{,\right\}_{1},
\notag\end{equation}
then $ m $ is a regular point of $ \varphi|_{C} $, i.e., $ \frac{d}{dt}\varphi|_{C}\not=0 $ at $ m $.
\item
The previous conditions on $ I $ are sufficient for $ I $ being a regular
point.
\end{enumerate}
\end{theorem}
\begin{proof} The first part of the theorem is trivial, since the
corresponding ideal is coming from $ S^{n'}M\times S^{n''}M $, and the bihamiltonian
structure is a direct product. Hence we can consider only the case when
the ideal $ I $ has support at one point $ m\in M $.
We are free to change the second Poisson structure $ \left\{,\right\}_{2} $ on $ M $ to the
linear combination $ \left\{,\right\}_{2}-\alpha\left\{,\right\}_{1} $, $ \alpha=\varphi\left(m\right) $, that is degenerate at $ m $. After
this change $ \varphi\left(m\right)=0 $. It is clear that in that case all eigenvalues of $ r $
are 0. Let us remind that all Jordan block of $ r $ appear by pairs.
Therefore the
condition of regularity is equivalent to the fact that the
recursion operator $ r $ has only $ 2 $-dimensional kernel (hence it has only
two Jordan blocks of size $ n $). By the corollary
this is equivalent to a fact that the operator of multiplication by $ \varphi\in A $
in $ \operatorname{Hom}_{A/I}\left(I/I^{2},A/I\right) $ has only $ 2 $-dimensional kernel, or that the operator
of multiplication by $ \varphi^{n-1} $ does not vanish. We want to show that
in this case there is an element $ g\in I $ such that $ m $ is a regular point of $ g $,
i.e., $ dg|_{m}\not=0 $ (we can take $ g $ as an equation of the curve $ C $).
There is a natural decreasing filtration in $ A/I $ consisting of
images of functions on $ M $ with increasing orders of zero at $ m $:
\begin{equation}
F_{k}\left(A/I\right)=m^{k}A/I,
\notag\end{equation}
here $ m $ is identified with a maximal ideal in $ A $. We can consider also
the filtration in $ I $:
\begin{equation}
F_{k}\left(I\right)=m^{k}\cap\text{I,}
\notag\end{equation}
We need only to prove that
$ F_{1}\left(I\right)\not=F_{2}\left(I\right) $, or what is the same, that $ \dim F_{1}\left(A/I\right)/F_{2}\left(A/I\right)<2 $. We claim
that $ \dim F_{k}\left(A/I\right)=n-k $.
Indeed, we know that multiplication by $ \varphi^{n-1} $ is non-zero in
$ \operatorname{Hom}_{A/I}\left(I/I^{2},A/I\right) $, hence $ \varphi^{n-1}\not=0 $ in $ A/I $. Therefore $ I $ is generated by $ \varphi^{n} $
and some element $ g $ with $ dg|_{m}\not=0 $, that proves the necessary conditions
of the theorem.
To prove the last part of the theorem we should only inverse the
previous discussion.
We
should show only that multiplication by $ x $ in $ \operatorname{Hom}_{A/I}\left(I/I^{2},A/I\right) $ has
two
Jordan blocks if $ I=\left(x^{n},y\right) $. An element $ \alpha $ in $ \operatorname{Hom}_{A/I}\left(I/I^{2},A/I\right) $ is uniquely
determined by $ \alpha\left(x^{n}\right) $ and $ \alpha\left(y\right) $, and these two elements of $ A/I $ can be
arbitrary.
Hence $ \operatorname{Hom}_{A/I}\left(I/I^{2},A/I\right) $ as $ A/I $-module is isomorphic to a direct sum of
two copies of $ A/I $, and $ x $ acts in $ A/I $ as a Jordan block.\end{proof}
\begin{remark} Let us compare this description with the above example of $ S^{2}M $.
On the latter Hilbert scheme any ideal corresponds to a pair of points or
to a double point on some curve on $ M $ (as
the description with the blow-up shows). Therefore the only condition of
the theorem is that this pair is not on one level set or this curve is
transversal to the level sets of the
ratio of two Poisson structures. It is easy to see that this condition
coincides with the condition $ \widetilde\alpha\not=0 $ from the above example. \end{remark}
\begin{remark} We see that in the case $ n=2 $ the condition that the ideal
corresponds to some curve is trivial. However, already in the case $ n=3 $
this is not so. On the $ 6 $-dimensional manifold $ S^{3}M $ there is a whole
$ 2 $-parametric
subset corresponding to $ 3 $-tuples of collided points on $ M $ such that
the collision was ``from different directions''. If this $ 3 $-tuple
collides in the point $ m\in M $ then the corresponding ideal consists of
functions on $ M $ with trivial $ 1 $-jet in $ m $, i.e., to
\begin{equation}
f\left(m\right)=0,\quad df|_{m}=0.
\notag\end{equation}
It is easy to see that this is exactly three conditions on a function $ f $.
Of course, any ideal of codimension 3 with support in $ m $ can be
described as a result of collision of three points on $ M $. If three points
were
moving along the same curve $ C $, then the corresponding ideal comes from
$ C $, as in the theorem. We can see that this is a generic case of an ideal
with support at $ m $: there is a $ 2 $-parametric family of such ideals. However,
if we cannot approximate the movement of
these three points by some common curve, then the resulting ideal is the
described above. Therefore we came to a very strange fact: a collision
in general position results in a special ideal, and some special
collisions results in ideals in general position.
\end{remark}
\begin{remark} Another consequence of the description of the bivector field in
terms of the ideal is a possibility to describe weak leaves of
codimension 2.
Consider an ideal $ I\in S^{n}M $ and a weak leaf $ L $ passing through the point $ I $. Let
$ L $ be a symplectic leaf for $ \left\{,\right\}_{1}-\lambda\left\{,\right\}_{2} $. We
can represent $ I $ as a product of relatively prime ideals $ I_{0} $ and $ I_{1} $ such
that
$ I_{0} $ has the support on the curve $ \varphi_{1}-\lambda\varphi_{2}=0 $, $ I_{1} $ has the support outside of this
curve. Now we can see that the above arguments have already proved the
following \end{remark}
\begin{proposition} We can compute $ \operatorname{codim} L $ as
\begin{equation}
\dim \operatorname{Ker} \mu_{\varphi_{1}-\lambda\varphi_{2}}\colon \operatorname{Hom}_{A/I_{0}}\left(I_{0}/I_{0}^{2},A/I_{0}\right) \to \operatorname{Hom}_{A/I_{0}}\left(I_{0}/I_{0}^{2},A/I_{0}\right).
\notag\end{equation}
Here we use the notations of the previous section. Therefore $ \operatorname{codim} L=2 $ is
equivalent to $ I_{0} $ being a regular point of $ S^{k}M $,
$ k=\dim A/I_{0} $. Therefore $ I_{0} $ has support in $ m_{0}\in M $ such that $ \varphi_{1}\left(m_{0}\right)=\lambda\varphi_{2}\left(m_{0}\right) $,
and the closure $ \overline L $ of $ L $ consists of ideals inside the maximal ideal $ I_{m_{0}} $.
\end{proposition}
\subsection{The compact case revisited }\label{s6.4}\myLabel{s6.4}\relax Now we have made all the preparations
for a look on the known
examples of compact bihamiltonian systems from the point of view of
classification theorems. Consider a Hilbert scheme of a compact surface
with two
global Poisson structures. We want to show that though not any point of
these manifold is a regular point, the weak classification theorem from
the section~\ref{s1.7} {\em is applicable\/} in any point of these manifolds.
Indeed, we know all the weak leaves of codimension 2 on $ S^{n}M $: a
closure of such a leaf consists of ideals that are supported in some maximal
ideal $ I_{m} $, $ m\in M $. Moreover, if $ n>1 $, then this point $ m $ can be any point but a
common zero
of two Poisson structures. Therefore, if $ M $ is connected and two Poisson
structures are linearly independent, then the closure of the incidence
set from the weak classification theorem coincides with the natural
incidence set
\begin{equation}
C''=\left\{\left(m,I\right) \mid I\subset I_{m}\right\}\subset M\times S^{n}M.
\notag\end{equation}
Now we only need to show that the natural projection $ C''\to S^{n}M $ is a flat
mapping, what is a standard fact of the theory of Hilbert schemes.
\begin{remark} Now we finished a circle in the description of the
bihamiltonian systems. First, in the section~\ref{s1.7} we showed that under
some mild conditions a point of a bihamiltonian manifold can be described
as a point on a Hilbert scheme of some canonically defined surface. Then
in the section~\ref{h5} we constructed examples of compact bihamiltonian
manifolds as Hilbert schemes of compact surfaces. At last, in this section we
show that these Hilbert schemes satisfy indeed the conditions of the
classification theorem. Therefore, first, we cannot weaken the conditions
of the weak classification theorem, and second, the conclusions are
sufficiently weak to be true on a compact manifold. \end{remark}
\begin{remark} Now we can also see how the generalized weak leaves look
like. They are of two different types: either closures of a weak leaf---%
i.e., the ideals in a given (generic) maximal ideal; or the ideals in the
maximal ideal such that both Poisson
structures vanish in the corresponding point. We see that in the case of
the Hilbert scheme of a plane with 9 blown-up points there is no leaves
of the second kind, but they do exist if we blow up lesser number of
points.
However, we can see that if we forget about Poisson structures,
both this types of generalized weak leaves look the same. Here we want to
consider an example of possible singularity on a closure of a weak leaf.
In fact what we are doing here is to investigate
\begin{equation}
L_{m}=\left\{I\in S^{n}M \mid I_{m}\supset I\right\}
\notag\end{equation}
for a fixed $ m\in M $.
\end{remark}
\begin{example} We have seen in the section~\ref{s7} that in the case $ n=2 $ the
submanifold $ L_{m} $ is smooth (and equal to the blow-up of $ M $ in $ m $). Let us
consider the case $ n=3 $.
It is easy to understand that the only point $ I_{0} $ on $ L_{m} $ that can be
singular is the result of a generic collision of a triple of points to $ m $.
We can consider a local frame such that $ m $ is a solution of $ x=y=0 $. The
corresponding ideal is $ \left(x^{2},xy,y^{2}\right) $. Consider a nearby ideal $ I $. It should
contain a function that is near to $ x^{2} $, the same for $ xy $ and $ y^{2} $. The
transversality allows us to correct these monoms by terms of the form $ Ax+By+C $
to get an element if $ I $. Let us denote the corresponding functions
\begin{align} x^{2} & +ax+b y+\alpha
\notag\\
xy & +cx+dy+\beta
\notag\\
y^{2} & +e x+fy+\gamma.
\notag\end{align}
From the other side, a tangent vector to $ S^{3}{\mathbb A}^{2} $ at $ I_{0} $ is a mapping
from
\begin{multline}\operatorname{Hom}_{{\mathbb C}\left[x,y\right]/\left(x^{2},xy,y^{2}\right)}\left(\left(x^{2},xy,y^{2}\right)/\left(x^{4},x^{3}y,\dots ,y^{4}\right),{\mathbb C}\left[x,y\right]/\left(x^{2},xy,y^{2}\right)\right)
\\
= \operatorname{Hom}_{{\mathbb C}\left[x,y\right]/\left(x,y\right)}\left(\left(x^{2},xy,y^{2}\right)/\left(x^{3},x^{2}y,xy^{2},y^{3}\right),\left(x,y\right)/\left(x^{2},xy,y^{2}\right)\right).
\notag\end{multline}
Therefore, if we denote $ T_{0}{\mathbb A}^{2} $ by $ V $, then this tangent space is just
$ \operatorname{Hom}_{{\mathbb C}}\left(S^{2}V,V\right) $. Hence the functions $ a,b,c,d,e,f $ form a good
coordinate system in a neighborhood of $ I_{0} $.
We want to find the equations of
the subset $ L_{0} $ of $ S^{3}{\mathbb A}^{2} $ consisting of the contained in $ \left(x,y\right) $ ideals. This
subset is of dimension 4, and it easy to see that $ I_{0} $ is a singular point
of this manifold. Indeed, the $ \operatorname{SL}\left(2\right) $-action shows that there is only one
invariant subspace of dimension 4 in the tangent space to $ S^{3}{\mathbb A}^{2} $ at $ I_{0} $, and
this subspace
obviously consists of triples with the center of mass at the origin.
(The complimentary $ 2 $-dimensional invariant space consists of
translations of $ I_{0} $.)
From the other side, the tangent cone to $ L_{0} $ at $ I_{0} $ is $ \operatorname{SL}\left(2\right) $-invariant,
therefore if it were smooth, it would conincide with that subspace, what
is obviously wrong.
However, it is not so difficult to write the equation for a tangent cone
to this subset at $ I_{0} $ explicitely.
Indeed, if $ \left(x,y\right)\supset I $, then $ \alpha=\beta=\gamma=0 $. We claim that there are 2 ways to
get a homogeneous element of degree 2 in $ I $. First, we can take a
linear combination of the above elements with a vanishing linear part,
what is
\begin{equation}
\left(cf- d e\right)x^{2}+\left(b e -af\right)xy+\left(a d -bc\right)y^{2}\in I.
\notag\end{equation}
Second, we can use the relation $ x^{2}\cdot y^{2}=\left(xy\right)^{2} $ and substitute instead of
quadratic monoms the congruent linear functions, what gives
\begin{equation}
\left(ae-c^{2}\right)x^{2}+\left(af+b e-2cd\right)xy+\left(bf-d^{2}\right)y^{2}\in I.
\notag\end{equation}
Compatibility gives us equations of $ L_{0} $:
\begin{equation}
\frac{cf- d e}{ae-c^{2}}=\frac{b e-af}{af+b e-2cd}=\frac{a d-bc}{bf-d^{2}}
\notag\end{equation}
(it is easy to see that these conditions are sufficient for the ideal
\begin{equation}
\left(x^{2}+ax+b y,xy+cx+dy,y^{2}+e x+fy\right)
\notag\end{equation}
to be of codimension 3).
We see that even in the simplest possible case the tangent cone in a
singular point is given by rather complicated equations. \end{example}
\begin{remark} We have seen in the previous remark that a generalized weak
leaf looks exactly as the closure of the weak leaf if we forget about the
Poisson structure on it. Therefore the singular points on it have the
same geometry. However, it is a union of weak leaves of codimension $ \geq4 $,
therefore it is interesting to investigate how these leaves are
positioned in a neighborhood of the singular point.
So suppose that the origin is a common zero for both Poisson
structures on $ {\mathbb A}^{2} $. Then the considered above subset
\begin{equation}
L_{0} = \left\{I\in S^{3}{\mathbb A}^{2} \mid I\subset\left(x,y\right)\right\}
\notag\end{equation}
is a generalized
weak leaf. A closure of a weak leaf of generic position inside $ L_{0} $ consists of
ideals
of codimension 3 inside the ideal $ \left(x,y\right)\cdot\left(x-x_{0},y-y_{0}\right) $, where $ x_{0}\not=0 $ or $ y_{0}\not=0 $.
The equations of this subset in the coordinates $ a,\dots ,f $ are
\begin{align} x_{0}^{2} +ax_{0}+b y_{0} & =0
\notag\\
x_{0}y_{0} +cx_{0}+dy_{0} & =0
\notag\\
y_{0}^{2} +e x_{0}+fy_{0} & =0,
\notag\end{align}
therefore these submanifolds are flat sections of the cone in question.
These sections miss the vertex of the cone, are flat and isomorphic to
the blow-up of the plane at the origin and at the point $ \left(x_{0},y_{0}\right) $.
The others weak leaves of dimension 2 are limits of the above ones
when the point $ \left(x_{0},y_{0}\right) $ goes to the origin. So consider the limit of
$ \left(\varepsilon x_{0},\varepsilon y_{0}\right) $ when $ \varepsilon\to $0. The corresponding equations in the coordinates
$ a,\dots ,f $ are
\begin{align} ax_{0}+b y_{0} & =0
\notag\\
cx_{0}+dy_{0} & =0
\notag\\
e x_{0}+fy_{0} & =0,
\notag\end{align}
we can suppose $ x_{0}=0 $. The equations of the weak leaf become $ b=d=f=0 $, and the
equation of the cone after this restriction become
\begin{equation}
ae-c^{2}=0.
\notag\end{equation}
We see that these $ 2 $-dimensional weak leaves (that are exeptions!) have
simpler singularities
than the $ 4 $-dimensional leaves (that correspond to the case of general
position!).
\end{remark}
\subsection{The Magri subset }\label{s6.5}\myLabel{s6.5}\relax Now, when we know the set of regular points in
$ S^{n}M $, we want to
show what this set is already described in the Magri work.
First, we suppose that $ M $ is $ {\mathbb C}^{2}=T^{*}{\mathbb A}^{1} $, the first Poisson structure is the
standard one $ \frac{d}{dx}\wedge\frac{d}{dy} $ and the second is
$ x\frac{d}{dx}\wedge\frac{d}{dy} $ (any generic bihamiltonian surface can be reduced locally to
such a form by a
coordinate change and a change $ \left\{,\right\}_{i}\mapsto\alpha_{i1}\left\{,\right\}_{1}+\alpha_{i2}\left\{,\right\}_{2} $ with some constant
$ \alpha_{ij} $). Here we are going to introduce the coordianate system on the
Hilbert scheme that establishes a connection between the subset of
regular points and the Magri coordinate system on a bihamiltonian
manifold in general position.
Consider the subset $ U $ of $ S^{n}M $ consisting of regular points
$ I $ on
$ S^{n}M $ such that the first Poisson structure is non-degenerate in these
points. Any such point satisfies the
following condition: if $ \left\{m_{1},m_{2},\dots ,m_{k}\right\} $ is the support of
the ideal $ I $, then all the $ x $-coordinates $ x\left(m_{1}\right),x\left(m_{2}\right),\dots ,x\left(m_{k}\right) $ of these
points are (finite and) distinct. In this case the factors $ I_{l} $, $ l=1,\dots ,k $,
of the ideal
$ I $ at points $ m_{l} $, $ l=1,\dots ,k $, determine some $ n_{l} $-jets of curves in these
points that
are transversal to the level sets $ x=\operatorname{const} $. That means that we can find a
curve $ C=\left\{\left(x,y\right) \mid y=f\left(x\right)\right\} $ with given jets in points $ m_{i} $.
Since
\begin{equation}
\sum_{l=1}^{k}\left(n_{l}+1\right)=n,
\notag\end{equation}
we can in fact choose $ f $ to be a polynomial of degree $ n-1 $, and this
condition determines the curve $ C $ in the unique way. We call this
polynomial $ f_{I} $.
If we know the curve $ C $, then to determine the ideal $ I $ it is sufficient
to find the corresponding ideal in the ring of functions on $ C $. (We should
remind that the ideal $ I $ is by definition a direct image of an ideal on
$ C $.) However, the projection $ x $ on $ {\mathbb A}^{1} $ identifies this ring with the
ring of functions of $ x $.
Any ideal in the ring of functions on line is uniquely determined by
its support (considered as a finite subset of $ {\mathbb C} $ with multiplicities). In
turn, this subset $ \left< x_{1},x_{2},\dots ,x_{n} \right> $ is uniquely determined by
the values of the elementary symmetric functions on it.
Here we want to show that {\em the Magri coordinate system is associated
with a particular choice of the set of symmetric functions,\/} with $ s_{l}=\sum
x_{i}^{l} $, $ l=1,\dots ,n $, This choice identifies $ S^{n}{\mathbb A}^{1} $ with a subset in the dual
space to the vector space of polynomial of degree $ \leq n $ by
\begin{equation}
\left(x_{1},\dots ,x_{n}\right)\mapsto\left(P\mapsto\sum_{i}P\left(x_{i}\right)\right).
\notag\end{equation}
Indeed, we see that to determine the ideal $ I\in U\subset S^{n}M $ it is sufficient to
provide the
corresponding polynomial $ f $ of degree $ n-1 $ in $ x $ and a linear functional
\begin{equation}
l_{I}\colon P\mapsto\sum_{i}P\left(x_{i}\right)
\notag\end{equation}
on the vector space $ {\mathcal P}_{n} $ of polynomials of degree $ n $. Since $ l_{I} $ sends 1 to $ n $,
it depends essentially only on the derivative
$ P'\in{\mathcal P}_{n-1} $:
\begin{equation}
l_{I}\left(P\right)=nP\left(x_{0}\right)+\widetilde l_{I}\left(P'\right).
\notag\end{equation}
Let us consider instead the corresponding functional on $ {\mathcal P}_{n-1} $:
\begin{equation}
\widetilde l_{I}\colon P\mapsto\sum_{i}\int_{x_{0}}^{x_{i}} P\left(t\right)dt.
\notag\end{equation}
A change of the constant $ x_{0} $ results only in an addition of an
independent of $ I $ functional on $ {\mathcal P}_{n-1} $, i.e., the translation of
the image of $ T^{*}{\mathbb A}^{1} $ in $ {\mathcal P}_{n-1}^{*} $, what is irrelevant in what follows. We put
$ x_{0}=0 $.
Hence we identified $ U $ with $ {\mathcal P}_{n-1}\times{\mathcal P}_{n-1}^{*}=T^{*}{\mathcal P}_{n-1}^{*} $. On the latter
vector space there is a natural symplectic $ 2 $-form
\begin{equation}
\left(\left(f_{1},\varphi_{1}\right),\left(f_{2},\varphi_{2}\right)\right) = \left< f_{1},\varphi_{2} \right> - \left< f_{2},\varphi_{1} \right>,
\label{equ6.3}\end{equation}\myLabel{equ6.3,}\relax
that determines a translation-invariant symplectic or Poisson
structure. Let us show that this structure coincides with the first
Poisson structure on $ S^{n}M=T^{*}{\mathbb A}^{1} $. It is sufficient to show this on an open dense
subset, hence we can consider the subset of $ U $ where all $ n $ points
$ x_{1},\dots ,x_{n} $ are different.
If $ n=1 $, then the ideal $ I $ (i.e., a point $ \left(x_{1},y_{1}\right)\in M $) goes to a
constant function $ f_{I}\left(x\right)=y_{1} $ and a functional $ \widetilde l_{I}\colon 1\mapsto x_{1} $, so the claim is
evident in this case. In general case let $ I $ correspond to $ \left\{\left(x_{i},y_{i}\right)\right\} $,
$ i=1,\dots ,n $, in $ \left(x,y\right) $-representation. We can represent any tangent vector
$ \left\{\left(\delta x_{i},\delta y_{i}\right)\right\} $ at $ I $ as $ \delta x_{i}=P\left(x_{i}\right) $, $ \delta y_{i}=Q\left(y_{i}\right) $ with appropriate $ P,Q\in{\mathcal P}_{n-1} $, and
the bracket of two such vectors with respect to the {\em symplectic\/} structure is
\begin{equation}
\left\{\left(P,Q\right),\left(\widetilde P,\widetilde Q\right)\right\}_{1}=\sum_{i}\left(P\widetilde Q-\widetilde PQ\right)\left(x_{i}\right).
\notag\end{equation}
On the other side, consider $ \left(f,\widetilde l\right) $-representation. Since $ f\left(x_{i}\right)=y_{i} $, the
$ {\mathcal P}_{n-1} $-component $ \delta f $ of this tangent vector is
\begin{equation}
Q-\sum_{i}P\left(x_{i}\right)f'\left(x_{i}\right)T_{i},
\notag\end{equation}
where $ T_{i} $ is the only polynomial of degree $ n-1 $ with zeros in $ x_{j} $, $ j\not=i $, and
a value 1 in $ x_{i} $. The $ {\mathcal P}_{n-1}^{*} $-component is
\begin{equation}
p\mapsto\sum_{i}P\left(x_{i}\right)\left(\left(\frac{d}{dx}\right)^{-1}p\right)'\left(x_{i}\right)=\sum_{i}P\left(x_{i}\right)p\left(x_{i}\right),
\notag\end{equation}
so the symplectic structures coincide indeed.
We can see now that we have identified an open subset $ U $ of the
set of regular points on $ S^{n}\left(T^{*}{\mathbb A}^{1}\right) $ with $ T^{*}\left(S^{n}{\mathbb A}^{1}\right) $ (here, in $ 1 $-dimensional
case, the Hilbert scheme $ S^{n}{\mathbb A}^{1} $ coincides with $ \left({\mathbb A}^{1}\right)^{n}/{\mathfrak S}_{n} $), and the first Poisson
structure on $ U $ goes to the natural Poisson structure on the cotangent
bundle.
It is also easy to see now that the second Poisson structure can be
also described easily in terms of $ {\mathcal P} $ and $ {\mathcal P}^{*} $. It is slightly easier to work
with symplectic structures again, so consider the open subset $ x_{i}\not=0 $, $ i=1,\dots ,n $,
where the
second Poisson structure is non-degenerate. Working with symplectic
structures allows as consider the pairing of tangent vectors instead of
cotangent, and the bracket of the
above tangent vectors with respect to this (second) symplectic
structure is
\begin{equation}
\left\{\left(P,Q\right),\left(\widetilde P,\widetilde Q\right)\right\}_{2}=\sum_{i}\left(P\widetilde Q-\widetilde PQ\right)\left(x_{i}\right)/x_{i}.
\notag\end{equation}
Therefore, if we denote by $ M_{\left\{x_{i}\right\}} $ a linear operator in the space\footnote{That is the vertical tangent space for the cotangent bundle to $ {\mathcal P}_{n-1}^{*} $.} $ {\mathcal P}_{n-1} $
such that
\begin{equation}
\left(M_{\left\{x_{i}\right\}}f\right)\left(x_{l}\right) =f\left(x_{l}\right)x_{l},\quad l=1,\dots ,n,
\notag\end{equation}
then the corresponding symplectic form
in $ {\mathcal P}_{n-1}\times{\mathcal P}_{n-1}^{*} $ at $ \left(f,\widetilde l\right) $, where $ \widetilde l $ corresponds to $ \left\{x_{i}\right\} $, is
\begin{equation}
\left(\left(\delta f_{1},\delta\varphi_{1}\right),\left(\delta f_{2},\delta\varphi_{2}\right)\right) = \left< M_{\left\{x_{i}\right\}}^{-1} \delta f_{1},\delta\varphi_{2} \right> - \left< M_{\left\{x_{i}\right\}}^{-1} \delta f_{2},\delta\varphi_{1} \right>.
\notag\end{equation}
That means that the corresponding Poisson pairing (given by the inverse
pairing matrix) can be written as
\begin{equation}
\left(\left(df_{1},d\varphi_{1}\right),\left(df_{2},d\varphi_{2}\right)\right) =\left< M_{\left\{x_{i}\right\}}^{*} df_{1},d\varphi_{2} \right> - \left< M_{\left\{x_{i}\right\}}^{*} df_{2},d\varphi_{1} \right>.
\label{equ6.6}\end{equation}\myLabel{equ6.6,}\relax
Here $ df_{1,2} $ are linear functionals on $ {\mathcal P}_{n-1}=T_{\widetilde l}^{*}{\mathcal P}_{n-1}^{*} $, $ d\varphi_{1,2} $ are linear
functionals on $ {\mathcal P}_{n-1}^{*} $.
Let us note now that $ M_{\left\{x_{i}\right\}}^{*} $ depends polynomially on the point
$ \widetilde l_{\left\{x_{i}\right\}}\in{\mathcal P}_{n-1}^{*} $.
Indeed, $ M_{\left\{x_{i}\right\}} $ is essentially the multiplication
by $ x $ corrected by a term killing the coefficient at $ x^{n} $:
\begin{equation}
M_{\left\{x_{i}\right\}}f=xf-\left(\text{the leading coefficient of $ f $}\right)\cdot P_{\left\{x_{i}\right\}},
\label{equ6.8}\end{equation}\myLabel{equ6.8,}\relax
where $ P_{\left\{x_{i}\right\}} $ is the only polynomial of degree $ n $ with the leading
coefficient 1 andzeros in $ x_{i} $, $ i=1,\dots ,n $. The coefficients of $ P_{\left\{x_{i}\right\}} $ are
another set of elementary symmetric functions of $ x_{i} $,
\begin{equation}
P_{\left\{x_{i}\right\}}\left(x\right) = x^{n}-\sigma_{1}x^{n-1}+\sigma_{2}x^{n-2}-\dots \pm\sigma_{n}.
\notag\end{equation}
However, the variables $ \sigma_{k} $ depend polynomially on the variables $ s_{l} $, i.e.,
coordinates in the vector space $ {\mathcal P}_{n-1}^{*} $, therefore the operator
$ M_{\left\{x_{i}\right\}} $ depends polynomially on the point $ \widetilde l_{\left\{x_{i}\right\}}\in{\mathcal P}_{n-1}^{*} $, therefore the
second Poisson structure on the linear space $ T^{*}\left(S^{n}{\mathbb A}^{1}\right)=T^{*}{\mathcal P}_{n-1}^{*} $
is polynomial.
Formally speaking, we proved that these formulae are true only on the
open dense subset $ x_{i}\not=0 $, however we can extend them anywhere by
continuity. We get
the fact that the identification of the open subset $ U $ of the
set of regular points on $ S^{n}\left(T^{*}{\mathbb A}^{1}\right) $ with the vector space $ T^{*}{\mathcal P}_{n-1}^{*} $
transforms the first Poisson structure into a constant one, and the
second Poisson structure into a polynomial Poisson structure. In the
following section we consider another coordinate system that will
simplify this situation yet further.
However, the formulae we get coincide literally with the formulae
for a bihamiltonian structure in the Magri's coordinate system. Let us
consider the Magri's hypothesis. He considered the characteristic
polynomial of the {\em recursion operator.\/} As we have seen, this polynomial is
an exact square. Consider a mapping from the bihamiltonian manifold to
the set of polynomials that sends a point to a square root of this
polynomial. We call this mapping the {\em Magri mapping.\/}\footnote{In fact the Magri considered a slightly different mapping: instead of
considering the coefficients of the square root, that are the elementary
symmetric functions $ \sigma_{i} $, he considered the symmetric functions $ s_{i} $, exactly
as we here. However, the conditions of submersion are equivalent for
these two mappings, so we permitted ourselves to interchange these two
mappings.} The Magri theorem
claims that if this mapping is a
submersion, then in an appropriate coordinate system there is a local
normal form of the Poisson structures on the manifold (that coincides\footnote{To see this we can note that $ s_{i} $ are exactly the local Hamiltonians
in the original Magri mapping.}
with the formulae~\eqref{equ6.3},~\eqref{equ6.6},~\eqref{equ6.8}). From these formulae (or the
formulae of the Magri's paper) we can see that any such a point is a
regular point
of bihamiltonian structure, and the non-degeneracy condition is
satisfied. This shows that in fact our conditions are equivalent to the
Magri's ones.
Therefore combining two local classification theorems, the Magri's
one and the our, we get the following
\begin{corollary} The following two conditions on a point on a bihamiltonian
manifold are equivalent:
\begin{enumerate}
\item
The point is a regular point, any weak leaf passing through it
intersects the
set of good points, and the first Poisson structure is locally non-degenerate;
\item
The first Poisson structure is locally non-degenerate and the Magri
mapping is a submersion.
\end{enumerate}
\end{corollary}
It is not clear how to get this corollary more directly.
\subsection{Another coordinate system }\label{s6.6}\myLabel{s6.6}\relax Here we want to show that if
instead of considering the
elementary symmetric functions $ s_{i} $ on $ S^{n}{\mathbb A}^{1} $ we consider the elementary
symmetric functions $ \sigma_{i} $, then the formulae of the previous section can be
simplified a lot. Considering a particular set of functions on $ S^{n}{\mathbb A}^{1} $ is
just a way to introduce a coordinate system on this set, so to rewrite
the formulae of the previous section in another coordinate system we want
first to give a coordinate-independent description of two Poisson
structures.
It is very easy with the first Poisson structure, since it is just a
canonical Poisson structure on $ T^{*}\left(S^{n}{\mathbb A}^{1}\right) $, therefore we can easily rewrite
it in any coordinate system. However the description~\eqref{equ6.6} of the second
Poisson structure uses the decomposition of the tangent space to a point
in $ T^{*}\left(S^{n}{\mathbb A}^{1}\right) $ into a horizontal and a vertical parts, what is much more
difficult to rewrite. Here we give another description of the second
Poisson structure on $ T^{*}\left(S^{n}{\mathbb A}^{1}\right) $.
In the previous section we defined an endomorphism $ M_{\left\{x_{i}\right\}} $ of the
linear space $ {\mathcal P}_{n-1} $, and considered it as an endomorphism of the cotangent
space to $ S^{n}{\mathbb A}^{1}={\mathcal P}_{n-1}^{*} $ at the point $ \left\{x_{i}\right\}\in S^{n}{\mathbb A}^{1} $. Here we
want to consider this family of
mapping of
cotangent spaces as a universal mapping $ M\colon T^{*}S^{n}{\mathbb A}^{1} \to T^{*}S^{n}{\mathbb A}^{1} $.
\begin{proposition} Consider a subset $ W=\left\{x_{i}\not=0 \mid i=1,\dots ,n\right\} $ of $ S^{n}{\mathbb A}^{1} $. The
restriction of $ M $ on $ T^{*}W $ is a diffeomorphism, and the second Poisson
structure on $ T^{*}W $ is a direct image of the first structure under
the action of this diffeomorphism:
\begin{equation}
\left\{\varphi_{1},\varphi_{2}\right\}_{2}=\left\{\varphi_{1}\circ M,\varphi_{2}\circ M\right\}_{1}.
\label{equ6.12}\end{equation}\myLabel{equ6.12,}\relax
\end{proposition}
\begin{proof} It is sufficient to prove this on an open dense subset of
configurations of different points. Locally on this subset $ S^{n}{\mathbb A}^{1} $ is
isomorphic to $ \left({\mathbb A}^{1}\right)^{n} $, so $ T^{*}\left(S^{n}{\mathbb A}^{1}\right) $ is locally isomorphic to a direct
product $ \left(T^{*}{\mathbb A}^{1}\right)^{n} $. Both Poisson structures, as well as the mapping $ M $ can be
written as direct products, so it is sufficient to consider the case $ n=1 $,
that is obvious. \end{proof}
\begin{remark} The fact that the second Poisson structure can be written by both
the formulae~\eqref{equ6.6} and~\eqref{equ6.12} requires very special properties of the
mapping
$ M $. These properties are insured by the following nice, simple, and
totally unexpected lemma that expresses symplectic properties of the
dependence of $ \sigma_{i} $ on $ s_{i} $. To formulate it we need to repeat some definitions.
Consider the coordinate system $ s_{i} $, $ i=1,\dots ,n $, on $ S^{n}{\mathbb A}^{1} $. It
essentially identifies
$ S^{n}{\mathbb A}^{1} $ with a dual space to the vector space of polynomials of degree $ \leq n $
with a zero at the origin:
\begin{equation}
\left\{x_{i}\right\}\mapsto\left(p\left(x\right)\mapsto\sum_{i}p\left(x_{i}\right)\right).
\notag\end{equation}
The differentiation identifies this space of polynomials with $ {\mathcal P}_{n-1} $.
Denote the corresponding mapping $ S^{n}{\mathbb A}^{1}\to{\mathcal P}_{n-1}^{*} $ by $ S $.
Consider the coordinate system $ \sigma_{i} $, $ i=1,\dots ,n $, on $ S^{n}{\mathbb A}^{1} $. It
essentially
identifies $ S^{n}{\mathbb A}^{1} $ with the space of polynomials of degree $ n $ with a leading
coefficient 1:
\begin{equation}
\left\{x_{i}\right\}\mapsto T\left(x\right),\quad T\left(x_{i}\right)=0\text{, }i=1,\dots ,n.
\notag\end{equation}
The translation by $ -x^{n} $ identifies the latter space with $ {\mathcal P}_{n-1} $. Denote the
corresponding mapping $ S^{n}{\mathbb A}^{1}\to{\mathcal P}_{n-1} $ by $ \Sigma $.
\end{remark}
\begin{lemma} Consider the mapping
\begin{equation}
S\times\Sigma\colon S^{n}{\mathbb A}^{1}\to{\mathcal P}_{n-1}^{*}\times{\mathcal P}_{n-1}=T^{*}{\mathcal P}_{n-1}^{*}.
\notag\end{equation}
The image of this mapping is a lagrangian submanifold, and the
corresponding $ 1 $-form on $ {\mathcal P}_{n-1}^{*} $ is $ -\frac{ds_{n+1}}{n+1} $. Here we consider
the elementary symmetric function $ s_{n+1} $ as a function of $ s_{1},\dots ,s_{n} $.
\end{lemma}
Now, when we know the coordinate-independent expressions for the
Poisson structures in question, we can write them down in the coordinate
system $ \sigma_{i} $ on $ S^{n}{\mathbb A}^{1} $. The only thing we need to do is to write down the
expression of the operator $ M $ in the new coordinate system.
\begin{lemma} Denote the dual to $ \sigma_{i} $ coordinates on $ T^{*}{\mathcal P}_{n-1} $ by $ \Sigma_{i} $. Then the
matrix of the operator $ M_{\left\{x_{i}\right\}} $ in this basis is
\begin{equation}
\left(
\begin{matrix}
\sigma_{1} & \sigma_{2} & \dots & \sigma_{n-1} & \sigma_{n}
\\
-1 & 0 & \dots & 0 & 0
\\
0 & -1 & \dots & 0 & 0
\\
\vdots & \vdots & \ddots & \vdots & \vdots
\\
0 & 0 & \dots & -1 & 0
\end{matrix}
\right)=M_{ij}.
\notag\end{equation}
The second Poisson structure can be written as
\begin{equation}
\left\{f,g\right\}_{2}=\sum_{ij} M_{ji} \left(\frac{\partial f}{\partial\sigma_{i}}\frac{\partial g}{\partial\Sigma_{j}}-\frac{\partial g}{\partial\sigma_{i}}\frac{\partial f}{\partial\Sigma_{j}}\right) + \sum_{ij} N_{ij}
\frac{\partial f}{\partial\Sigma_{i}}\frac{\partial g}{\partial\Sigma_{j}},
\notag\end{equation}
where
\begin{equation}
N_{ij}=\left(
\begin{matrix}
0 & \Sigma_{2} & \dots & \Sigma_{n}
\\
-\Sigma_{2} & 0 & \dots & 0
\\
\vdots & \vdots & \ddots & \vdots
\\
-\Sigma_{n} & 0 & \dots & 0
\end{matrix}
\right).
\notag\end{equation}
\end{lemma}
We see that in this coordinate system two Poisson brackets in
question are of the simplest possible form: the first is constant, the
second is linear. In fact we defined a pair of affine Poisson brackets on
$ T^{*}{\mathcal P}_{n-1} $.
\subsection{The corresponding Lie algebra }\label{s6.7}\myLabel{s6.7}\relax Let us remind the usual description of
affine Poisson brackets:
\begin{lemma} Call a Poisson bracket on a linear space $ V $ an affine bracket, if
the bracket of two linear functions is a linear (nonhomogeneous)
function. There is a $ 1-1 $ correspondence between affine Poisson brackets
on $ V $ and pairs $ \left(\left[,\right],c\right) $, where $ \left[,\right] $ is a structure of Lie algebra on
$ V^{*} $, and $ c $ is a $ 2 $-cocycle on $ V^{*} $. \end{lemma}
\begin{proof} Define the Lie operation on $ V^{*} $ as a linear part of the Poisson
bracket:
\begin{equation}
\left[\varphi_{1},\varphi_{2}\right]=\text{the linear part of $ \left\{\varphi_{1},\varphi_{2}\right\} $},
\notag\end{equation}
and the cocycle as
\begin{equation}
c\left(\varphi_{1},\varphi_{2}\right)= \left\{\varphi_{1},\varphi_{2}\right\}|_{0}.
\notag\end{equation}
The inverse operation is the consideration of the corresponding to $ c $
central extension $ \widetilde V^{*} $ of $ V^{*} $, and the identification of $ V $ and the subspace
of $ \left(\widetilde V^{*}\right)^{*} $ passing through $ c\in\left(\widetilde V^{*}\right)^{*} $. Due to this identification the Lie---%
Kirillov bracket on $ \left(\widetilde V^{*}\right)^{*} $ defines a Poisson bracket on $ V $.\end{proof}
The formulae of the previous section show that any linear
combination
\begin{equation}
\lambda\left\{,\right\}_{1}+\left\{,\right\}_{2}
\notag\end{equation}
of brackets on $ T^{*}\left(S^{n}{\mathbb A}^{1}\right) $ is affine in the coordinate
system $ \sigma_{i} $, and the linear parts of these brackets coincide. That means
that on the dual space to $ T^{*}{\mathcal P}_{n-1} $ there is a structure of Lie algebra.
Moreover, there are two cocycles $ c_{1} $, $ c_{2} $ for this algebra, and the
Poisson brackets $ \lambda\left\{,\right\}_{1}+\left\{,\right\}_{2} $ are associated with the sums $ \lambda c_{1}+c_{2} $.
To write down this Lie algebra structure let me remind that we write
a generic polynomial $ p\in{\mathcal P}_{n-1} $ as
\begin{equation}
-\sigma_{1}x^{n-1}+\sigma_{2}x^{n-2}-\dots \pm\sigma_{n}
\notag\end{equation}
(so $ \sigma_{i} $ are linear coordinate functions on $ {\mathcal P}_{n-1} $, $ \sigma_{i}\in{\mathcal P}_{n-1}^{*} $), and we call the
dual coordinates on $ {\mathcal P}_{n-1}^{*} $ by $ \Sigma_{i} $, $ \Sigma_{i}\in{\mathcal P}_{n-1} $ (in fact $ \Sigma_{i}=\left(-1\right)^{i}x^{n-i} $).
\begin{lemma} The only non-zero brackets of basic elements for the Lie algebra
structure on
$ {\mathcal P}_{n-1}\times{\mathcal P}_{n-1}^{*} $ associated with the second Poisson structure are
\begin{equation}
\left[\Sigma_{1},\Sigma_{i}\right]=\Sigma_{i}\text{, }i\not=1,\quad \text{and\quad }\left[\Sigma_{1},\sigma_{k}\right]=-\sigma_{k}.
\notag\end{equation}
The only non-zero coordinates of the cocycle $ c_{1} $ are
\begin{equation}
c_{1}\left(\sigma_{i},\Sigma_{i}\right)=1,\quad i=1,\dots ,n.
\notag\end{equation}
The only non-zero coordinates of the cocycle $ c_{2} $ are
\begin{equation}
c_{2}\left(\sigma_{i},\Sigma_{i+1}\right)=1,\quad i=1,\dots ,n-1.
\notag\end{equation}
\end{lemma}
\begin{remark} It is interesting to find some algebraic conditions on this
algebra that make it appear in this geometrical situation. It is easy to
recognize the Jordan case of undecomposable pairs of bilinear forms in
this pair of cocycles. One conjectural description would be that this is
a generic case of a Lie algebra structure on a vector space such that
this pair of forms is a pair of cocycles. See the next section for a
discussion of a simplest example $ n=2 $.
In that section we show that in
this particular case the set of compatible Lie algebra structures has two
irreducible components of maximum dimension, and that any of these
components contains an open orbit of the group of automorphisms of the
pair of forms. Moreover, though two Lie algebra structures
corresponding to these components are
noon-isomorphic, the corresponding local bihamiltonian structures {\em are\/}
isomorphic (so this isomorphism is non-linear). We will see also that the
considered here structure corresponds to one of these two components
indeed. \end{remark}
\subsection{Examples of linearizations and non-smooth spaces $ M^{\left(2\right)} $ }\label{s6.8}\myLabel{s6.8}\relax The
discussion
in the previous section allows as to formulate the following
\begin{problem} Consider a pair of skewsymmetric bilinear forms $ \alpha $, $ \beta $ in a vector
space $ V $.
Find all Lie algebra structures in $ V $ such that $ \alpha $ and $ \beta $ are cocycles. \end{problem}
We have seen that such a structure determines a bihamiltonian
structure in the space $ V^{*} $. It is especially interesting to consider this
problem in the case when $ \alpha $ and $ \beta $ form an undecomposable pair of forms. In
this section we give the solution of this problem in the first
non-trivial case, when $ \dim V=4 $ and $ \alpha $, $ \beta $ form a pair that corresponds to
a Jordan block.
\begin{theorem} \label{th6.2}\myLabel{th6.2}\relax Consider a pair of skew forms
\begin{equation}
a^{*}\wedge\alpha^{*}+b^{*}\wedge\beta^{*},\quad a^{*}\wedge b^{*}
\notag\end{equation}
in $ 4 $-dimensional vector space $ V $ with a basis $ a,b,\alpha,\beta $. These
forms are $ 2 $-cocycles for the following Lie algebras:
\begin{enumerate}
\item
$ \left[\alpha,\beta\right]=\beta $, $ \left[\alpha,a\right]=a $, $ \left[\alpha,b\right]=-b $;
\item
$ \left[\alpha,\beta\right]=2\beta $, $ \left[\alpha,a\right]=a $, $ \left[\alpha,b\right]=-b $, $ \left[\beta,b\right]=a $;
\item
$ \left[a,\alpha\right]=\alpha $, $ \left[b,\beta\right]=\beta $;
\item
$ \left[a,\alpha\right]=\beta $, $ \left[b,\alpha\right]=\alpha $, $ \left[b,\beta\right]=\beta $;
\item
$ \left[b,\beta\right]=\beta $;
\item
$ \left[b,\alpha\right]=\beta $;
\item
$ \left[\bullet,\bullet\right]=0 $.
\end{enumerate}
In this list we write only non-zero brackets. Moreover, any Lie
algebra structure for which these forms are cocycles can be transformed
to one of these forms by a linear transformation of $ V $ which preserves
this pair of forms.
\end{theorem}
\begin{remark} Consider the set $ {\mathfrak L} $ of all Lie algebra structures on $ V $ such that
the above forms are cocycles. The theorem claims that the group $ G $ of
automorphisms of this pair of forms acts on $ {\mathfrak L} $ with 7 orbits. Denote by $ H $
the subgroup of $ G $ consisting of elements which preserve vectors $ \alpha $ and $ \beta $.
Note that $ G=\operatorname{SL}_{2} \ltimes H $. Then
the first two orbits are principal homogeneous spaces for $ G $, the
stabilizer of the third is $ {\mathbb Z}/2{\mathbb Z} \ltimes H $, the stabilizers of the fourth
and fifth are $ H $, the stabilizer of the sixth is $ {\mathbb Z}/3{\mathbb Z} \ltimes H $, and the
seventh orbit consists of one point. Here the generator of $ {\mathbb Z}/2{\mathbb Z} $ is the
element
\begin{equation}
\alpha\mapsto-\beta,\quad \beta\mapsto\alpha,\quad a\mapsto-b,\quad b\mapsto a,
\notag\end{equation}
of $ \operatorname{SL}_{2} $, and the generator of $ {\mathbb Z}/3{\mathbb Z} $ is the element
\begin{equation}
\alpha\mapsto\mu\alpha,\quad \beta\mapsto\mu^{-1}\beta,\quad a\mapsto\mu^{-1}a,\quad b\mapsto\mu b,
\notag\end{equation}
here $ \mu^{3}=1 $.
It is interesting also to understand which of these orbits are
adjacent. Unfortunately, the simple analysis leading to the theorem
~\ref{th6.2} could not give the answer on this question. A cumbersome and
absolutely straightforward calculation shows that the picture is as the
following:
\begin{equation}
\begin{matrix}
\left(1\right) & & \left(2\right) & & \left(3\right)
\\
& \searrow & \downarrow
\\
\left(4\right) & \to & \left(6\right) & \to & \left(7\right)
\\
& & \uparrow
\\
& & \left(5\right)
\end{matrix}
\notag\end{equation}
However, it is unclear how to check that this calculation contains no
error, so one should handle this statement with some care. One check is
the compatibility with the classification of bihamiltonian systems. The
description below shows that there is no immediate contradiction with the
geometric intuition.
If we accept the above statement, we can see that $ {\mathfrak L} $ contains 5
irreducible components in two connected components. The
bihamiltonian structure considered in the previous section corresponds to
Lie algebra structure that is a point in an open subset of one
of two irreducible components of maximal dimension. We will see that the
points in another orbit of maximal dimension correspond to the same
(local) bihamiltonian structure.
\end{remark}
It is easy to understand how to write the Poisson
bracket $ \left\{,\right\}_{\lambda} = \lambda\left\{,\right\}_{1}+\left\{,\right\}_{2} $ that corresponds to any particular case of
the theorem. We
want to investigate this bracket in the third case of the theorem.
\begin{example} On $ V^{*} $ we can consider coordinates $ a,b,\alpha,\beta $, and the basic
brackets are:
\begin{equation}
\left\{a,\alpha\right\}_{\lambda}=\alpha+\lambda,\quad \left\{b,\beta\right\}_{\lambda}=\beta+\lambda,\quad \left\{a,b\right\}_{\lambda}=1.
\notag\end{equation}
The conditions that two cocycles form a Jordan pair imply that the origin
is a regular point on $ V^{*} $ with a double eigenvalue 0. Consider the space
of weak leaves.
The Pfaffian of the corresponding to $ \left\{,\right\}_{\lambda} $ bivector is $ \left(\alpha+\lambda\right)\left(\beta+\lambda\right) $,
therefore this bivector is degenerate in two cases: $ \alpha=-\lambda $ and $ \beta=-\lambda $. Let $ \alpha=-\lambda $.
Under
this restriction a bracket of $ \alpha $ or $ e^{a}\left(\beta-\alpha\right) $ with any other function is 0.
Therefore
\begin{equation}
\alpha=\alpha_{0},\quad e^{a}\left(\beta-\alpha\right) =\beta_{0}
\notag\end{equation}
are equations of weak leaves. In the same way $ \beta=-\lambda $ gives the second set
of weak leaves:
\begin{equation}
e^{-b}\left(\beta-\alpha\right)=\alpha_{1},\quad \beta =\beta_{1}.
\notag\end{equation}
However, if $ \beta_{0}=\alpha_{1}=\beta_{1}-\alpha_{0}=0 $, then these two families of equations give
the same weak leaf. Hence the parameter space of weak leaves is a
union of two planes intersecting by a line.
{\em Therefore we get an example of a regular bihamiltonian structure that
has a
non-smooth parameter space of weak leaves!\/} Two Poisson structures on this
space are given by
\begin{gather} \left\{\alpha_{0},\beta_{0}\right\}_{1}=-\alpha_{0}\beta_{0},\quad \left\{\beta_{1},\alpha_{1}\right\}_{1}=\beta_{1}\alpha_{1}.
\notag\\
\left\{\alpha_{0},\beta_{0}\right\}_{2}=-\beta_{0},\quad \left\{\beta_{1},\alpha_{1}\right\}_{2}=\alpha_{1}.
\notag\end{gather}
Any one of these two Poisson structures corresponds to bivector fields on
the intersecting planes. We can see that the bivector fields on these
planes vanish on the intersection line with opposite linear parts.
\end{example}
\begin{remark} Let us list the descriptions of bihamiltonian structures in the
remaining examples. The first two examples lead to the same bihamiltonian
structure as the considered in the previous section (for $ \dim =4 $). The first one
leads
to the same coordinate system as before, the second one to a different
coordinate
system. The remarkable property of the latter coordinate system is the
fact that not only one of the Poisson structures is constant and the
other one affine, but also the Lie derivative by a constant vector field
$ \frac{\partial}{\partial a} $ transforms the linear one into the constant one! We do not know if
it is possible to do the same in the case $ \dim \geq6 $.
We have already considered the third case. In the fourth case we get
an example of a bihamiltonian structure with bivector fields forming a
Jordan pair at any point. Turiel introduced the multidimensional
generalization of this example in the paper \cite{Tur89Cla}.
In the fifth example we get again a pair a planes intersecting by a
line as a parameter space of weak leaves. However, in this case one
Poisson structure is as above, the
other Poisson structure vanishes on one plane, and on the second one it
has a zero of the second order on the intersection line.
In the sixth example the pair of Poisson structure can be
transformed to a translation-invariant form in the coordinate system
$ a,b-a^{2}/2,\beta,\beta a+\alpha $. In
the seventh example the pair of forms is already translation-invariant.
\end{remark}
\begin{remark} We want to explain here how to interpret the above example of
non-smooth $ M^{\left(2\right)} $ using the language of Hilbert schemes. If $ M^{\left(2\right)} $
{\em is\/} smooth, then $ M $ can be identified with a piece of the Hilbert scheme
$ S^{k}M^{\left(2\right)} $ (here $ k=2 $). We want to analyze here what can be a possible
generalization of
this fact to a case of non-smooth manifold. We know already both $ M $ and
$ M^{\left(2\right)} $,
below we compute $ S^{2}M^{\left(2\right)} $ and see that $ S^{2}M^{\left(2\right)} $ is non-smooth, but normal,
and $ M $ is an irreducible component of the {\em normalization\/} of $ S^{2}M^{\left(2\right)} $.
Consider
a union $ M $ of two planes $ \pi_{1} $ and $ \pi_{2} $ in $ 3 $-dimensional (projective) space.
Consider a Hilbert scheme of this variety on the level 2.
Consider three open subsets on $ S^{2}M $: the first consists of pairs of
different points, one on each of planes; and the other two consist of
pairs of different points on either one of planes. The closures of
these open subsets form three irreducible components of $ S^{2}M $. The last two
components are clearly smooth. We are going to study the geometry of the
remaining component.
To
any pair of different points on $ M $ we can associate a line passing through
these points. It is easy to see that we can extend this mapping to a
mapping from the Hilbert scheme to the set of lines in the space. The
preimage of a line consists of one point of the Hilbert scheme excepting
the case when this line is inside $ M $. In the latter case the preimage
is a $ 1 $-dimensional manifold naturally identified with the line in
question, except the case when this line is $ \pi_{1}\cap\pi_{2} $, when this preimage is
the symmetric square of this line.
Since two subsets $ \Pi_{1} $, $ \Pi_{2} $ consisting of lines inside $ \pi_{i} $ intersect
transversely in the set of lines in the space, we can consider a blow-up
$ L $ of the latter space in these two subvarieties. The order of two blow-ups
is irrelevant because of the transversality. The preimage of $ \Pi_{1}\smallsetminus\Pi_{2} $
consists of lines in $ \pi_{1} $ with a marked point, the same for $ \Pi_{2}\smallsetminus\Pi_{1} $, the
preimage
$ \Pi_{12} $ of $ \Pi_{1}\cap\Pi_{2} $ consists of ordered subsets of two points on $ \Pi_{1}\cap\Pi_{2} $.
We see that the first irreducible component of $ S^{2}M $ can be identified
with the quotient of $ L $ by the action of the symmetric group $ {\mathfrak S}_{2} $ on the
submanifold $ \Pi_{12} $. The only non-smooth points on this quotient are the
points on the image of $ \Pi_{12} $. Consider a point on this image. In a local
coordinate system $ \Pi_{12} $ is given by the equations $ x=y=0 $ and $ {\mathfrak S}_{2} $ is acting by
$ \left(0,0,z,t\right)\mapsto\left(0,0,-z,t\right) $. We can split off the variable $ t $ and consider the
$ 3 $-dimensional manifold with coordinates $ x,y,z $ and an action of $ {\mathfrak S}_{2} $ on
$ x=y=0 $ by $ \left(0,0,z\right)\mapsto\left(0,0,-z\right) $.
The basic coordinate functions on the quotient are $ \left(x,y,xz,yz,z^{2}\right) $.
We can see that $ \sqrt{z^{2}} = \frac{xz}{x} $ is an element in both the integer closure
and the field of ratios of this ring, therefore the {\em normalization\/} of the
quotient is the initial $ 3 $-dimensional space.\footnote{The analogous $ 2 $-dimensional example where $ {\mathfrak S}_{2} $ is acting on $ x=0 $ as
$ \left(0,y\right)\mapsto\left(0,-y\right) $ corresponds to the famous {\em Whitney's umbrella}. Indeed, the
basic coordinate functions on the quotient are
\begin{equation}
a=x,\quad b=xy,\quad c=y^{2},
\notag\end{equation}
and the relation is $ b^{2}-a^{2}c=0 $.
\endgraf
In this example it is easy to draw the
corresponding picture and to see that the result of the normalization
(i.e., of the separating of two intersecting sheets of the umbrella) is
the initial plane.} Hence {\em the normalization of
the Hilbert scheme is smooth}. One of three connected components of this
normalization is the discussed above blow-up of the space of lines.
Now to a Poisson structure on $ M $ we can associate a Poisson structure
on an open subset of $ S^{2}M $. However, we cannot apply the proof from the
section~\ref{s7} to extend this Poisson structure to the whole $ S^{2}M $: there
are
additional hypersurfaces where the corresponding bivector field can
have a pole. They are two exceptional divisors on the blow-up. In fact a
simple calculation shows that this bivector field {\em has\/} a pole unless the
bivector fields on the components of $ M $ have opposite linear parts on the
intersection. (These bivector field {\em should\/} vanish on the intersection for
the Poisson bracket of two functions to be a function on $ M $.) \end{remark}
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