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\topmatter
\title Webs, Veroneze curves and bihamiltonian systems \endtitle
\author Israel M.~Gelfand, Ilya S.~Zakharevich \endauthor
\date January 1991 \enddate
\address Beloosersky lab, The Moscow State University, Leninskie
gory, Moscow, 117234, USSR \endaddress
\address The Institute for Problems in
Mechanics, pr.~Vernadskogo,
101, Moscow, 117526, USSR \endaddress
\abstract We define a special kind of multidimensional webs,
connected with the Veroneze curve. For these webs the foliations in
question depend not on a discrete parameter, but on the point on a
projective line. For each bihamiltonian system of odd
dimension in general position we construct such a web and show how
to reconstruct the original bihamiltonian system basing on these data.
\endabstract
\toc
%\head{D.}
\head{} Definitions of basic objects \endhead
\subhead{} Veroneze webs\endsubhead
\subsubhead{\quad} Veroneze curves \endsubsubhead
\subsubhead{\quad} The main definition of a Veroneze web \endsubsubhead
\subsubhead{\quad} Equivalent definitions \endsubsubhead
\subsubhead{\quad} Examples \endsubsubhead
\subhead{} Bihamiltonian structures \endsubhead
\head{} Intermediate theorems \endhead
\subhead{} Linear algebra connected with a pair of skew-symmetrical forms \endsubhead
\subhead{} The geometry of a Poisson structure \endsubhead
\head{} Two maps \endhead
\subhead{} Reduction of a bihamiltonian structure \endsubhead
\subhead{} A reverse map \endsubhead
\head{} Double complex \endhead
\subhead{} The bicomplex \endsubhead
\subhead{} Objects of differential geometry \endsubhead
\subhead{} Double cohomology \endsubhead
\head{} Twist of a bihamiltonian structure \endhead
\head{} Geometry of a bihamiltonian structure \endhead
\subhead{} An affine structure \endsubhead
\subhead{} Connection with lagrangian foliations \endsubhead
\subhead{} Translations and Schouten---Nijenhuis brackets \endsubhead
\head{} Double cohomology of a double complex \endhead
\head{} Cited literature \endhead
\endtoc
\endtopmatter
\document
\head Definitions of basic objects \endhead
\subhead Veroneze webs\endsubhead Let us fix a two-dimensional linear space $ {\Cal S} $ and call
it the fundamental space\footnote{We have chosen this letter (although there appears a visual conflict
between two S's in the notation $ S^{k}{\Cal S} $ below) to accent the apparent
connection of this space with a spinor space. The action of the
special linear group for the space $ {\Cal S} $ (denoted as $ \operatorname{SL}\left({\Cal S}\right) $) will be one of main
tools in the discussion below.}. By $ PV $ we denote the
projectivization
of a vector space $ V $, i.e., the space of $ 1 $-dimensional subspaces in
$ V $.
\subsubhead Veroneze curves \endsubsubhead Since the notion of a Veroneze curve will play
a crucial r\^ole in what follows, we discuss it up to subtleties here.
\definition{Definition} A {\it Veroneze inclusion\/} in $ k $-dimensional projective
space is a
mapping $ P{\Cal S}\hookrightarrow PV $ such that there exists an isomorphism $ PV\to PS^{k}{\Cal S} $ which
makes the diagram
$$
\CD P{\Cal S} @>>> PV
\\
@| @VVV
\\
P{\Cal S} \SubSup!R{S^{k}\operatorname{id}} PS^{k}{\Cal S} \endCD
$$
commutative.\enddefinition
Here $ S^{k}V $ is the $ k $-th symmetrical power of $ V $,
$ S^{k}\varphi $ for a map $ \varphi\colon V\to W $ denotes the corresponding map
$ V\to S^{k}W\colon v\mapsto\varphi\left(v\right)^{k} $ or the corresponding map of projectivizations.
\example{Example } The more habitual definition of a Veroneze curve is the
curve that in a coordinate system is parameterized as $ \left(t,t^{2},\dots ,t^{k}\right) $.
So a Veroneze curve on a line is the line itself, on a plane is a
parabola (or any conic). The connection between this two definitions
is established by the choice
$$
{\Cal S}=''\text{the space of homogeneous linear functions in }\left(y_{1},y_{2}\right)''.
$$
In this case
$$
S^{k}{\Cal S}=''\text{the space of homogeneous polynomials of degree }k\text{ in
}\left(y_{1},y_{2}\right)''.
$$
It is clear that $ S^{k}\operatorname{id} $ sends a linear function $ \left(t_{1}y_{1}+t_{2}y_{2}\right)\in{\Cal S} $
to a
polynomial $ \left(t_{1}y_{1}+t_{2}y_{2}\right)^{k}\in S^{k}{\Cal S} $. Multiplication of this linear function
by a constant results in multiplication of this polynomial by
another constant, so this map can be reduced to a map of
projectivizations.
So $ \left(y_{1}+ty_{2}\right) $ goes to
$$
\left(y_{1}+ty_{2}\right)^{k} = 1\cdot\binomSS_{0}^{k}y_{1}^{k}+t\cdot\binomSS_{1}^{k}y_{1}^{k-1}y_{2}+t^{2}\cdot\binomSS_{2}^{k}y_{1}^{k-2}y_{2}^{2}+\dots +t^{k}\cdot\binomSS_{k}^{k}y_{2}^{k},
$$
and this vector-valued function in $ t $ differs from the function
$ t\mapsto\left(t,t^{2},\dots ,t^{k}\right)=\left(1:t:t^{2}:\dots :t^{k}\right) $ only by a coordinate change.
Hence any curve that in some coordinate system is parameterized
by $ \left(t,t^{2},\dots ,t^{k}\right) $ is a Veroneze curve in the sense of our definition.
\endexample
\example{Example } Another useful parameterization of a Veroneze curve is
$$
t\mapsto\left(\left(t-\lambda_{1}\right)^{k},\left(t-\lambda_{2}\right)^{k},\dots ,\left(t-\lambda_{k}\right)^{k}\right).
$$
To show that the curve $ P{\Cal S}\to PS^{k}{\Cal S} $ allows this parameterization let us
chose the ``interpolation coordinate system'' on the space of
homogeneous polynomials of degree $ k $:
$$
p\left(y\right)\mapsto\left(p\left(1,0\right),p\left(-\lambda_{1},1\right),p\left(-\lambda_{2},1\right),\dots ,p\left(-\lambda_{k},1\right)\right).
$$
Clearly, the curve $ \left(y_{1}+ty_{2}\right)^{k} $ has in this coordinate system
the wanted parameterization.
So any curve with such a parameterization is a Veroneze curve.
\endexample
\remark{Remark } In fact any curve of degree $ k $ in $ k $-dimensional space that
isn't contained in any hyperspace is a Veroneze curve (we mean that
it is rational---i.e., the projective line if considered as a manifold---%
and any
identification of this curve with $ P{\Cal S} $ is a Veroneze inclusion). \endremark
\remark{Remark } We have chosen this awkward definition of a Veroneze curve
since it opens a possibility to speak as invariantly as it is
possible. In fact the identification $ \alpha\colon PV\to PS^{k}{\Cal S} $ from the definition is
uniquely defined. Indeed, the inverse image of the sheaf $ {\Cal O}\left(1\right) $ on $ PV $
is the sheaf $ \alpha^{*}{\Cal O}\left(1\right)\simeq
\left(S^{k}\operatorname{id}\right)^{*}{\Cal O}\left(1\right)=
{\Cal O}\left(k\right) $. Hence a linear function on $ V $ (i.e.,
a global section of $ {\Cal O}\left(1\right) $) corresponds to a function on $ {\Cal S}^{*} $ of
homogeneity degree $ k $ after a choice of the last isomorphism.
However, this isomorphism is a section of $ {\Cal O}\left(k\right)\otimes{\Cal O}\left(k\right)^{*}={\Cal O}\left(0\right) $. So
isomorphism between $ \alpha^{*}{\Cal O}\left(1\right) $ and $ {\Cal O}\left(k\right) $ is defined up to a constant,
hence the isomorphism between $ V^{*} $ and $ S^{k}{\Cal S}^{*}=\left(S^{k}{\Cal S}\right)^{*} $ is defined up to a
constant. \endremark
\remark{Remark } The previous remark shows that for any Veroneze inclusion
$ \alpha\colon P{\Cal S}\to PV $ we can define an action of a projective linear group $ \operatorname{PSL}\left({\Cal S}\right) $
on the space $ V $. Really,
denote the corresponding identification of $ PV $ and $ PS^{k}{\Cal S} $ by $ i_{\alpha} $. If
$ g\in\operatorname{PSL}\left({\Cal S}\right) $, then $ \alpha\circ g $ is another Veroneze inclusion. So we can consider
another identification $ i_{\alpha\circ g} $ of $ PV $ and $ PS^{k}{\Cal S} $. These two
identifications
differ by a map $ \beta_{g}=i_{\alpha\circ g}^{-1}\circ i_{\alpha} $ in $ PV. $ We can consider $ g\mapsto\beta_{g} $ as a
projective action of $ \operatorname{PSL}\left({\Cal S}\right) $ on $ V $.
It is known, however, that any
such action can be uniquely pushed up to a linear action of a special
linear group of $ {\Cal S} $ (called $ \operatorname{SL}\left({\Cal S}\right) $) on
$ V $. It is easy to see that another definition of this action is the
inverse image of $ \operatorname{SL}\left({\Cal S}\right) $-action on $ S^{k}{\Cal S} $ under an identification of $ V $
and $ S^{k}{\Cal S} $. Therefore this action is irreducible. \endremark
\proclaim{ Proposition } To determine a Veroneze inclusion in
the space
$ PV $ it is sufficient to determine an irreducible $ \operatorname{SL}\left({\Cal S}\right) $-structure on
$ V $. \endproclaim
\demo{ Proof } Really, the image of a point $ \lambda\in P{\Cal S} $ in $ PV $, i.e., an
one-dimensional
subspace in $ V $, can be found as a highest-weight vectors subspace
relative to the Borel subgroup $ B_{\lambda}=\operatorname{Stab}\lambda $. \qed\enddemo
\subsubhead The main definition of a Veroneze web \endsubsubhead A Veroneze web can be
defined in many different ways. Here we begin with a definition that
will be used throughout {\it this\/} paper.
\definition{Definition} A {\it foliation\/} $ {\Cal F} $ of codimension $ l $ on a manifold $ X $ is a family of
subspaces $ {\Cal F}_{x}\subset T_{x}X $ of codimension $ l $ (for $ x\in X $) such that in an
appropriate coordinate system $ \left(x^{i}\right) $, $ i=1,\dots ,n $, on $ X $ the subspace $ {\Cal F}_{x} $
is generated by vector fields $ \partial/\partial x^{i} $, $ i=1,\dots ,n-l $. A submanifold $ L\subset X $
is called a {\it leaf\/} of the foliation $ {\Cal F} $ if $ T_{x}L={\Cal F}_{x} $ for $ x\in L $ and $ L $ is
a maximal connected submanifold with this property. A {\it tangent space to a
foliation\/} $ {\Cal F} $ in a point $ x\in X $ is by definition the space $ {\Cal F}_{x} $, a {\it cotangent
space\/} is
$ {\Cal F}_{x}^{*} $. A {\it tangent\/} (or {\it cotangent\/}) bundle to a foliation is linear bundle on
$ X $ with corresponding fibers. \enddefinition
Sometimes we will introduce a foliation by the set of it's
leaves.
As usual, for linear subspace $ W\subset V $ the subspace $ W^{\perp}\subset V^{*} $ is the
orthogonal complement to $ W $.
\remark{Remark } Let us consider a foliation $ {\Cal F} $ of codimension 1 on
$ X $. By
definition the leaves of this foliation can be determined as
$ f=\operatorname{const} $, $ f $ being a function on $ X $. So $ {\Cal F}_{x}=\left(df\right)^{\perp}=\left(g\,df\right)^{\perp} $ for an arbitrary
non-zero function $ g $. Note that if $ \omega=g\,df $, then $ \omega\wedge d\omega=0 $.
Inversely, let $ \omega $ be an $ 1 $-form on $ X $ such that
$$
\omega\wedge d\omega=0.
$$
It is known that we can find (locally) a function $ \varphi $ on $ X $ such that $ d\left(\varphi\omega\right)=0 $.
So $ \omega=\varphi^{-1}df $, and $ {\Cal F}_{x}=\left(\omega\right)^{\perp} $ is a foliation with the leaves $ \left\{f=\operatorname{const}\right\} $. \endremark
\definition{Definition} A {\it Veroneze web\/} on a $ \left(k+1\right) $-dimensional manifold $ X $ is
a family of codimension 1 foliations $ {\Cal F}_{\lambda} $ parameterized by $ \lambda\in P{\Cal S} $
such that in every point $ x\in X $ the orthogonal complements $ {\Cal F}_{\lambda,x}^{\perp} $ to
leaves
of foliations $ {\Cal F}_{\lambda} $ at $ x $ correspond to Veroneze inclusion $ P{\Cal S}\hookrightarrow PT_{x}^{*}X $,
$ \lambda\mapsto{\Cal F}_{\lambda,x}^{\perp} $. \enddefinition
\subsubhead Equivalent definitions \endsubsubhead The definition from the previous
section being useful in the connection with bihamiltonian
structures, in other domains there can be useful different
definitions of this object.
\proclaim{ Proposition } For a given Veroneze web on $ X $ there exists (locally)
a family of $ 1 $-forms $ \omega_{\lambda} $ on $ X $, $ \lambda\in{\Cal S} $, such that $ {\Cal F}_{\lambda,x}=\omega_{\lambda}\left(x\right)^{\perp} $ and
$$
\omega_{\lambda}\wedge d\omega_{\lambda}=0,
$$
these forms are homogeneous polynomials in $ \lambda\in{\Cal S} $ of degree $ k $ and
are non-degenerate in the sense that for any vector $ v\in T_{x}X\quad \left\not\equiv 0 $ in
$ \lambda $. Inversely, such a family on a manifold $ X $ determines (uniquely)
a Veroneze
web on $ X $. Two families differing only on multiplication by function
$ \varphi $ not
dependent on $ \lambda $
$$
\widetilde\omega_{\lambda}\left(x\right)=\varphi\left(x\right)\omega_{\lambda}\left(x\right),
$$
determine the same Veroneze web. \endproclaim
\demo{ Proof } The only non-evident assertion is the polynomial
dependence of $ \omega_{\lambda} $ in $ \lambda $. The polynomial map $ S^{k}\operatorname{id}\colon {\Cal S}\to S^{k}{\Cal S} $ sends $ {\Cal S} $ onto
the preimage of the Veroneze curve under the projection $ S^{k}{\Cal S}\to PS^{k}{\Cal S} $.
Taking an identification of $ S^{k}{\Cal S} $ and $ T_{x}^{*}X $ that depends smoothly in $ x $
we get what is claimed. \qed\enddemo
\proclaim{ Theorem } Consider $ 2k+1 $ foliations $ {\Cal F}^{i} $ of codimension 1 on $ X $
such that
\roster
\item
The orthogonal complements $ {\Cal F}^{i\perp} $ to the foliations considered as
points on $ PT_{x}^{*}X $ are different;
\item
for any
two points $ x,y\in X $ there exists a projective map $ PT_{x}^{*}X \to PT_{y}^{*}X $
sending this
set of $ 2k+1 $ points on $ PT_{x}^{*}X $ to the
corresponding set for the point $ y; $\footnote{We can express this by the phrase that {\it the configurations\/} formed by
these points on $ PT_{x}^{*}X $ and on $ PT_{y}^{*}Y $ are isomorphic.}
\item
and that in one (or in any)
point $ x\in X $ this set lies on an appropriate Veroneze curve.
\endroster
Then there exists (uniquely determined) Veroneze web $ {\Cal F}_{\lambda} $, $ \lambda\in{\Cal S} $, on $ X $ and
$ 2n+1 $ points $ \lambda_{m} $ on $ P{\Cal S} $ such that $ {\Cal F}^{m}={\Cal F}_{\lambda_{m}} $. These objects are determined up to
a simultaneous projective map $ P{\Cal S}\to P{\Cal S} $. \endproclaim
\demo{ Proof } It is easy to see that the Veroneze curve containing
the points
in question is uniquely determined. Therefore it is possible to
construct in this case a family of $ 1 $-forms $ \omega_{\lambda} $ that depends polynomially
in $ \lambda\in{\Cal S} $ and sweeps for any $ x\in X $ the this Veroneze
curve. The expression
$$
\omega_{\lambda}\wedge d\omega_{\lambda}
$$
is 0 as an homogeneous polynomial of degree $ 2k $ vanishing in $ 2k+1 $
points on a projective line
(since for $ 2n+1 $ different values of $ \lambda $ the form $ \omega_{\lambda} $ determines one of
the original $ 2k+1 $ foliations).\qed\enddemo
\subsubhead Examples \endsubsubhead In what follows we will use only the first of
these examples. However, the second shows that the object we have
introduces it quite classical.
\example{Example } Let $ i\colon P{\Cal S}\hookrightarrow PV^{*} $ be a Veroneze inclusion. For $ \lambda\in P{\Cal S} $ consider
a foliation on $ V $ with hyperplane leaves orthogonal to $ i\left(\lambda\right) $. It is
clear that this family of foliations form a Veroneze web. A {\it flat\/}
Veroneze web is a web that is diffeomorphic to this one.
\endexample
\example{Example } Consider the previous theorem in the case $ k=1 $. The only nontrivial
condition on foliation is the first one---the condition of
transversality. Hence any 3 transversal families of curves on a
plane determine a Veroneze web.
Let us rephrase the theorem in this special case. Fix three points
$ \lambda_{1},\lambda_{2},\lambda_{3}\in P{\Cal S} $. Let $ {\Cal F}_{\lambda_{1}} $, $ {\Cal F}_{\lambda_{2}} $, $ {\Cal F}_{\lambda_{3}} $ be
three arbitrary foliations (i.e., families of curves) on
two-dimensional manifold $ X $. Then $ {\Cal F}_{\lambda_{1},x}^{\perp},{\Cal F}_{\lambda_{2},x}^{\perp},{\Cal F}_{\lambda_{3},x}^{\perp}\in PT_{x}^{*}X $ (being three
points on a projective line) determine a projective coordinate
system on $ PT_{x}^{*}X $, i.e., an identification $ i_{x}\colon P{\Cal S}\to PT_{x}^{*}X $. For any $ \lambda\in P{\Cal S} $ the
distribution of lines $ i_{x}\left(\lambda\right)^{\perp}\subset T_{x}X $, $ x\in X $, is integrable (as any
distribution of lines), therefore it defines a Veroneze web. \endexample
\remark{Remark } The latter example shows that the ordinary
definition of a web (as of 3
families of curves) is just a special case of our definition. In
subsequent papers we are going to show that this
transition to continuous
families of foliations from a discrete family (that consist of three
foliations) is very
natural. In particular, the $ 6 $-angle invariant of the web obtained from
Veroneze web by choosing three points on $ P{\Cal S} $ {\it does not depend\/}
essentially on the choice of these points. It is this connection
with ordinary web that motivated us to choose this name. \endremark
\remark{Remark } We can see that on two-dimensional manifolds there exist
non-flat Veroneze webs (e.g., webs on a plane with
non-vanishing $ 6 $-angle
invariants). The interest of this definition is connected in
particular with existence of non-flat Veroneze webs for higher
dimensions and the rich geometry of these objects. \endremark
\subhead Bihamiltonian structures \endsubhead A bihamiltonian structure is a
pair of Poisson structures with an additional condition. The
definition of a Poisson structure is well known (see, e.g., \cite{6}).
\definition{Definition} A Poisson structure on a manifold $ Y $ is a bivector field
$ \eta\in\Gamma\left(\Lambda^{2}TY\right) $ such that a skew-symmetrical bracket
$$
\left\{,\right\}\colon \left(f,g\right)\mapsto\left\{f,g\right\},\qquad \left\{f,g\right\}\left(y\right)=\left<\eta\left(y\right),\left(df\wedge dg\right)\left(y\right)\right>
$$
satisfy the Jacoby identity
$$
\left\{\left\{f,g\right\},h\right\}+\left\{\left\{h,f\right\},g\right\}+\left\{\left\{g,h\right\},f\right\}=0.
$$
Here $ f $, $ g $, $ h $ (and $ \left\{f,g\right\} $) are functions on $ Y $.
\enddefinition
\definition{Definition} A
bihamiltonian
structure on a manifold $ Y $ is a pair of Poisson structures $ \eta_{1} $,
$ \eta_{2}\in\Gamma\left(\Lambda^{2}TY\right) $ such that any linear combination $ \mu_{1}\eta_{1}+\mu_{2}\eta_{2} $ is also a Poisson
structure. \enddefinition
Let us recall that in the tensor notations $ t_{mn kl,i} $ denotes the
derivative of the component $ t_{mn kl} $ of the tensor field $ t $ in the direction of
$ i $.
\remark{Remark } Let us note that the condition imposed on a bivector $ \eta $ to
be a Poisson structure
(in a local coordinate frame $ y^{i} $ on $ Y $ it can be written as
$$
\operatornamewithlimits{Alt}_{i j k}\eta^{il}\eta_{,l}^{jk}=0\text{)}
$$
is quadratic in $ \eta $. (Here $ \operatornamewithlimits{Alt}_{i j k} $ denotes the alternation operation:
$$
\operatornamewithlimits{Alt}_{i_{1}i_{2}\dots i_{l}}X_{i_{1}i_{2}\dots i_{l}}=\sum_{\sigma\in{\goth S}_{l}}\left(-1\right)^{\sigma}X_{\sigma\left(i_{1}\right)\sigma\left(i_{2}\right)\dots \sigma\left(i_{l}\right)},
$$
where $ {\goth S}_{l} $ is a symmetrical group.) Hence an
arbitrary linear combination of $ \eta_{1} $, $ \eta_{2} $
will be Poisson provided some fixed non-trivial combination (for
example, $ \eta_{1}+\eta_{2} $) is Poisson. \endremark
\definition{Definition} Denote by $ \left[\eta,\eta'\right] $ the vector-valued symmetrical
bilinear form
$$
\eta,\eta'\in\Gamma\left(\Lambda^{2}TY\right)\mapsto\left[\eta,\eta'\right]\in\Gamma\left(\Lambda^{3}TY\right)
$$
that corresponds to the quadratic form
$$
\eta\mapsto\left[\eta,\eta\right]=\operatornamewithlimits{Alt}_{i j k}\eta^{il}\eta_{,l}^{jk}.
$$
This is a canonically defined map $ S^{2}\left(\Gamma\left(\Lambda^{2}TY\right)\right)\to\Gamma\left(\Lambda^{3}TY\right) $, which is
called {\it Schouten\/}---{\it Nijenhuis bracket}. \enddefinition
\remark{Remark } For convenience in what follows we will understand the
bihamiltonian structure as two-dimensional linear family of
Poisson structures
parameterized by the fundamental space $ {\Cal S} $, $ \eta_{\cdot}\colon {\Cal S}\mapsto\Gamma\left(\Lambda^{2}TY\right) $. The connection
with previous
(ordinary) definition arises after a choice of a basis in $ {\Cal S} $,
$$
\eta_{\left(\lambda_{1},\lambda_{2}\right)}=\lambda_{1}\eta_{1}+\lambda_{2}\eta_{2},\qquad \left(\lambda_{1},\lambda_{2}\right)\in{\Cal S}.
$$
\endremark
The interest of bihamiltonian structure lies in the theory of
integrable systems, where a wide family of such structures arises
in a natural way \cite{1}. Inversely, the bihamiltonian structure can
produce an ``integrable system''\footnote{We use here quotation marks since the notion of integrable system
is too vague to denote something explicit.} by the Lennard scheme \cite{1}, which
makes it possible to construct sufficiently many Hamiltonians in
involution.
The geometry of a bihamiltonian structure has been deeply
investigated in the works of Magri \cite{2} and Gelfand---Dorfman \cite{3}.
Unfortunately, these considerations were based on the hypothesis that
one of the $ 2 $-vector\footnote{I.e., the element of 2nd skew-symmetrical power.} fields $ \eta_{1} $, $ \eta_{2} $ (say, $ \eta_{1} $) is
non-degenerated.
However, in a lot of interesting examples (as in the one of the simplest
one---the periodical Korteweg---de Vries system!) any linear
combination of this two Poisson
structures is degenerate. The linear algebra arising in
{\it one\/} cotangent space in such a case was examined in \cite{4}. It differs
radically from the situation in the non-degenerated case and resembles
greatly the situation in an odd-dimensional space. (It is clear that in
the last case {\it any\/} $ 2 $-vector is degenerate.)
In this example we consider a bivector in $ T_{y}Y $ as a
skewsymmetric
bilinear form on $ T_{y}^{*}Y $. In the case of Korteweg---de Vries system $ Y $ is the
space $ V $ of functions with the period 1. We will identify the space
$ T_{y}^{*}Y $ with the same space using the scalar product
$$
\left(f,g\right)\mapsto\int_{0}^{1}f\left(x\right)g\left(x\right)\,dx.
$$
After these preparations we can write two corresponding
bilinear forms as
$$
\align \left(f,g\right) \relSS{\mapsto}^{\eta_{1}} & \int_{0}^{1}f\left(x\right)\left(-\partial^{3}+2u\left(x\right)\circ\partial+2\partial\circ u\left(x\right)\right)g\left(x\right)\,dx,
\\
\left(f,g\right) \relSS{\mapsto}^{\eta_{2}}4 & \int_{0}^{1}f\left(x\right)\partial g\left(x\right)\,dx,\qquad \partial=\partial/\partial x.\endalign
$$
Any linear combination of these forms is degenerate, since
$$
\left(-\partial^{3}+2u\left(x\right)\circ\partial+2\partial\circ u\left(x\right)\right)\phi_{1}\phi_{2}=0
$$
if $ \phi_{1} $, $ \phi_{2} $ are solutions of $ \left(-\partial^{2}+u\left(x\right)\right)\phi\left(x\right)=0 $, and $ \phi_{1}\phi_{2} $ is periodic if
$ \phi_{1} $, $ \phi_{2} $ are two Bloch solutions of this equation.
Here we will make the first step in studying of the differential
geometry of a bihamiltonian structure on an odd-dimensional manifold. But
first let us recall some simple facts concerning the linear algebra
of two skew-symmetrical forms and the geometry of a Poisson manifold.
\head Intermediate theorems \endhead
\subhead Linear algebra connected with a pair of skew-symmetrical forms \endsubhead
Let $ \Omega_{1} $, $ \Omega_{2} $ be a pair of skew-symmetrical forms on a
finite-dimensional vector space $ V $.
\proclaim{ Theorem \cite{4} } In the above situation
\roster
\item
the triple $ \left(V,\Omega_{1},\Omega_{2} \right) $ can be transformed by a linear change $ V\simeq W\oplus\widetilde W^{*} $ to
the triple
$$
\left(W\oplus\widetilde W^{*},\widetilde\Omega_{1},\widetilde\Omega_{2} \right)
$$
where the $ 2 $-forms $ \widetilde\Omega_{1} $, $ \widetilde\Omega_{2} $ can be written as
$$
\widetilde\Omega_{i}\left(\left(w_{1},w_{1}^{*}\right),\left(w_{2},w_{2}^{*}\right)\right) = \left<\varphi_{i} \left(w_{1} \right),w_{2}^{*} \right> - \left<\varphi_{i 2}\left(w \right),w_{1}^{*} \right>,
$$
and $ \varphi_{1} $, $ \varphi_{2} $ are two maps $ W\to\widetilde W $, $ w_{1},w_{2}\in W_{1} $, $ w_{1}^{*},w_{2}^{*}\in\widetilde W^{*} $, the brackets
$ \left<,\right> $ denoting the pairing between $ \widetilde W $ and $ \widetilde W^{*} $.
\item
Moreover, let the space $ V $ be of an odd dimension $ 2k+1 $ and the forms $ \Omega_{1} $, $ \Omega_{2} $
be in general position. Then we can choose the maps $ \varphi_{1} $, $ \varphi_{2} $ as
follows:
$$
\gather \varphi_{1}\left(w_{i}\right)=\widetilde w_{i+1/2},\quad i0,\qquad \varphi_{2}\left(w_{0}\right)=0 \endgather
$$
in the bases $ w_{i} $, $ i=0,1,\dots ,k $, $ \widetilde w_{j} $, $ j=1/2,3/2,\dots ,k-1/2 $, of spaces
$ W $, $ \widetilde W $ (i.e., such pairs of form have no parameters up to coordinate
change).
\item
The subspace $ W\subset V $ in the last case is canonically defined and the map
$$
i\colon \left(l_{1}:l_{2}\right)\mapsto \operatorname{Ker}\left(l_{1}\Omega_{1}+l_{2}\Omega_{2}\right)\subset W
$$
considered as a map $ P^{1}\to PW $ is a Veroneze inclusion;
\item
The map
$$
j\colon \left(l_{1}:l_{2}\right)\mapsto\varphi_{1}\left(\operatorname{Ker}\left(l_{1}\Omega_{1}+l_{2}\Omega_{2}\right)\right)=\varphi_{2}\left(\operatorname{Ker}\left(l_{1}\Omega_{1}+l_{2}\Omega_{2}\right) \right)\subset\widetilde W
$$
considered as a map $ P^{1}\to P\widetilde W $ (if $ l_{1}=0 $ then only the second expresion
has sense, if $ l_{2}=0 $ only the second) is a Veroneze inclusion;
\item
hence the subspace $ W $ is a linear subspace generated by
$ \operatorname{Ker}\left(\lambda_{1}\Omega_{1}+\lambda_{2}\Omega_{2}\right)\subset W $ for $ \left(\lambda_{1}:\lambda_{2}\right)\in P^{1} $.
\endroster
\endproclaim
\remark{Remark } The example of a pair of mappings from part (2) of the
theorem is nothing else as a {\it Kroneker pair\/} of operators from one
space to another \cite{7}. {\it All the results\/} of this paper are based in
fact on a careful study of this pair. \endremark
\remark{Remark } Note, how far is this situation from that in an
even-dimensional space in general position, where the form $ \Omega_{1} $ is
non-degenerated, and $ A=\Omega_{1}^{-1}\Omega_{2} $ is the canonically defined map $ V\to V $ in
general position\footnote{Among maps with the double spectrum.}. All the invariants of this pair are the
eigenvalues of {\it A}. On the contrary, in the odd-dimensional case there
is no invariant at all! \endremark
\remark{Remark } As it is easy to see from the description of Veroneze
inclusions as $ \operatorname{SL}\left({\Cal S}\right) $-modules (here $ \operatorname{SL}\left({\Cal S}\right) $ denotes a special linear
group of $ {\Cal S} $), the another description of maps $ \varphi_{1,2} $
can be obtained from considering the only $ \operatorname{SL}\left({\Cal S}\right) $-invariant map
$$
S^{k}{\Cal S}\otimes{\Cal S}\to S^{k-1}{\Cal S}.
$$
\endremark
\subhead The geometry of a Poisson structure \endsubhead In what follows we will use
only two facts from the geometry of a Poisson or symplectic manifold:
the existence of a foliation with symplectic leaves and the geometry
of a lagrangian foliation.
\example{Example } Let $ \left(Y,\omega\right) $ be a symplectic manifold, $ \omega\in\Gamma\left(\Lambda^{2}T^{*}Y\right) $. Then we
can consider $ \omega_{y} $ as a map $ T_{y}Y\to T_{y}^{*}Y\colon v\mapsto\omega_{y}\left(v,\cdot\right) $. Analogously the map
$ \omega_{y}^{-1}\colon T_{y}^{*}Y\to T_{y}Y $ can be considered as a bivector from $ \Lambda^{2}T_{y}Y $. We claim
that this bivector field on $ Y $ is a Poisson structure. \endexample
Inversely, let us consider a Poisson manifold $ \left(Y,\eta\right) $. Then
on an open
subset of $ Y $ where $ \operatorname{rk} \eta $ is constant there exists a foliation $ {\Cal F} $ of
dimension $ \operatorname{rk} \eta $ and a symplectic structure $ \omega_{L} $ on any leaf $ L $ of $ {\Cal F} $
such that
$ i_{*}\left(\omega_{L}^{-1}\right)=\eta|_{L} $. Here $ \omega_{L}^{-1}\in\Gamma\left(\Lambda^{2}TL\right) $, $ i_{*} $ is a map associated with the
inclusion $ i:
L\hookrightarrow Y $, $ |_{L} $ sends $ \Gamma\left(Y,\Lambda^{2}TY\right) $ into $ \Gamma\left(L,i^{*}\left(\Lambda^{2}TY\right)\right) $. That means, in
particular, that the Poisson bracket associated with $ \eta $
$$
\left\{f,g\right\}=\eta\left(df,dg\right)
$$
can be reduced to the Poisson bracket on leaves of $ {\Cal F} $:
$$
\left\{f,g\right\}\left(y\right)=\left\{f\circ i_{y},g\circ i_{y}\right\}_{L_{y}}\left(y\right);
$$
here $ i_{y}\colon L_{y}\hookrightarrow Y $ is the inclusion of the containing $ y $ leaf $ L_{y} $, $ \left\{\cdot,\cdot\right\}_{L_{y}} $
is the Poisson structure on the symplectic manifold $ L_{y} $.
In what follows we will consider a family of Poisson
structures. To denote a
dependence of the foliation $ {\Cal F} $ on this variable structure we will denote it as
$ {\Cal F}_{\eta} $.
\remark{Remark } Let us show how to construct this foliation.
The bivector $ \eta_{y} $ can be considered as
a skew-symmetrical form on
the cotangent space $ T_{y}^{*}Y $. Therefore $ \operatorname{Ker} \eta_{y}\subset T_{y}^{*}Y $.
The conormal
bundle to this foliation is nothing else but the field of kernels of
the $ 2 $-vector field $ \eta $.
Being
of constant rang
on an open subset, this
field of subspaces is therefore a bundle.
Orthogonal complements to these spaces form a distribution $ \widetilde{\Cal F} $ of
subspaces in $ TY. $ To prove that this distribution is integrable
(i.e., of the form $ T{\Cal F} $) it is sufficient to apply the Frobenius
theorem. Indeed, if $ v_{1},v_{2}\in\widetilde{\Cal F}_{y} $, then there exist two functions $ f_{1} $, $ f_{2} $
such that
$$
v_{i}=\widetilde\eta_{y}\left(df_{i}|_{y}\right),\quad i=1,2.
$$
Here $ \widetilde\eta_{y} $ is a mapping
$$
\widetilde\eta_{y}\colon T_{y}^{*}Y\to T_{y}Y,\qquad \left<\widetilde\eta_{y}\left(\omega_{1}\right),\omega_{2}\right>=\eta_{y}\left(\omega_{1},\omega_{2}\right),\quad \omega_{1},\omega_{2}\in T_{y}^{*}Y.
$$
However, $ \operatorname{Im}\eta_{z}=\left(\operatorname{Ker}\eta_{z}\right)^{\perp} $, so vector fields
$ v_{i}\left(z\right)=\widetilde\eta_{z}\left(df_{i}|_{z}\right) $,
$ i=1,2 $ are tangent to the distribution $ \widetilde{\Cal F} $, and the commutator of
these fields
$$
\left\{v_{1},v_{2}\right\}|_{y}=\widetilde\eta_{y}\left(d\left\{f_{1},f_{2}\right\}|_{y}\right)\tag $ * $
$$
is also tangent to $ \widetilde{\Cal F} $, therefore the Frobenius theorem is
applicable\footnote{The formula (*), that establishes the correspondence between the
Poisson bracket on functions and the commutator of vector fields, is
an easy reformulation of the Jacoby identity.}. \endremark
A submanifold $ L $ of a symplectic manifold $ \left(Y,\omega\right) $ is called a {\it lagrangian
submanifold\/} if $ \omega|_{TL}=0 $, $ 2\dim L=\dim Y $. A foliation $ {\Cal F} $ on a symplectic manifold
is called a
{\it lagrangian foliation\/} if any leaf of $ {\Cal F} $ is a lagrangian submanifold.
One of the main reasons to study lagrangian foliation is to
find what additional information is contained in the symplectic
structure on conormal manifold $ Y=T^{*}X $ comparing with a generic
symplectic manifold. The obvious additional structure is the
projection $ T^{*}X\to X $ with lagrangian fibers and the zero section $ X\to T^{*}X $.
It appears that this is almost all the additional information.
\proclaim{ Theorem } Let $ {\Cal F} $ be a lagrangian foliation on $ Y $. Then
\roster
\item
on leaves of
$ {\Cal F} $ a canonical affine structure (i.e., a local identification
with an open subset
in an affine space) is defined.
\item
If $ \widetilde L $ is a lagrangian submanifold of
$ Y $ locally transversal to $ {\Cal F} $, then some neighborhood $ U $ of $ \widetilde L $ is
canonically identified with an open subset of $ T^{*}\widetilde L $. This
identification preserves the symplectic structures. The leaves of the
foliation are identified with the fibers of this bundle, the affine
structures being the same.
\item
If a map $ f\colon Y\to Y $ preserves any leaf of the foliation $ {\Cal F} $, then
the restriction of $ f $ on a leaf is a translation in the corresponding
affine structure, provided that $ f $ preserves the symplectic
form $ \omega $. If $ \widetilde L $ is a transversal to $ {\Cal F} $ lagrangian submanifold, then
the vectors of the translations can be considered as a section of
$ T^{*}\widetilde L $
(see previous assertion of this theorem). For a leafwise translation
$ f $ to preserve the symplectic structure this section should be closed
as an $ 1 $-form on $ \widetilde L $.
\endroster
\endproclaim
\demo{ Proof } Although this theorem is standard in symplectic geometry,
we give here some hints about it's proof. The reason is that in what
follows there appears a lot more complicated version of the same
arguments, so it is useful to discuss them in an easier case beforehand.
Let $ B $ be a {\it local base\/} of the foliation $ {\Cal F} $. We mean that $ Y $ can be
represented locally as a direct product $ Y=F\times B $ with leaves of $ {\Cal F} $ being
the fibers of the projection on the second argument. Let $ \pi\colon Y\to B $ be
a corresponding projection. Then the condition of the foliation
being lagrangian implies that the form $ \omega $ considered as a map $ T_{y}Y\to T_{y}^{*}Y $
sends $ T_{y}{\Cal F} $ to $ T_{\pi\left(y\right)}^{*}B $ isomorphically. Hence for $ b\in B $ all the tangent
spaces to a leaf $ L=\pi^{-1}\left(b\right) $ of $ {\Cal F} $ are identified. Hence, on $ L $ is
defined a flat connection $ \nabla $.
To show that this connection has no torsion, consider two
covectors $ \alpha_{1},\alpha_{2}\in T_{b}^{*}B $. Then for $ y\in L $ the map $ \omega_{y}^{-1}\colon T_{b}^{*}B\to T_{y}L $ sends
them to vectors $ v_{1}\left(y\right) $, $ v_{2}\left(y\right) $, that form two vector fields on $ L $.
These fields being constant respective to $ \nabla $, their commutator
corresponds to the value of the torsion on these vector fields.
To compute this commutator, we use again the correspondence (*) between
the Poisson bracket on functions and the commutator of vector
fields. Let $ \varphi_{1} $, $ \varphi_{2} $ be two functions on $ B $ satisfying
$$
d\varphi_{i}|_{b}=\alpha_{i},\quad i=1,2.
$$
Then for $ y\in L $
$$
\left[v_{1},v_{2}\right]|_{y}=\omega_{y}^{-1}\left\{\pi\circ\varphi_{1},\pi\circ\varphi_{2}\right\}|_{y}.
$$
Applying again the condition of the foliation being lagrangian, we
find out that the last Poisson bracket is 0. Hence the connection $ \nabla $
determines an affine structure on $ L $. Evidently, the associated
vector space coincides with $ T_{b}^{*}B $.
The choice of a transversal submanifold marks a
point in an affine space of any leaf $ L $. Hence taking this point as 0 we can
identify locally $ L $ and $ T_{x}^{*}\widetilde L $, $ x $ being $ L\cap\widetilde L $. We used a
natural identification of $ \widetilde L $ and $ B $. So locally $ Y $ is identified with
$ T^{*}\widetilde L $.
Since $ \widetilde L $ is a lagrangian submanifold, the symplectic forms on $ Y $
and $ T^{*}\widetilde L $
coincide in the points of $ \widetilde L $. Really, for arbitrary transversal in any point
they coincide (being both 0)
on tangent to $ {\Cal F} $ (or fibers) vectors, coincide by construction of an
affine structure on pairs of vectors one of which is tangent to $ {\Cal F} $
(indeed, we can change the second vector to a tangent to $ B=\widetilde L $
vector). Moreover, in points of $ \widetilde L $ they coincide (being both 0) on
tangent to $ \widetilde L $ (or the zero section) vectors, what proves the
assertion.
Now we can proliferate this coincidence to any point of $ Y $ using
{\it Hamiltonian flows}. Let $ f $ be a function on a symplectic manifold $ Z $. Then the
vector field $ \omega^{-1}\left(df\right) $ preserves the form $ \omega $:
$$
{\Cal L}_{\omega^{-1}\left(df\right)}\cdot\omega=0.
$$
Therefore the {\it flow\/} $ T^{t} $ of this vector field also preserves $ \omega $:
$$
\left(T^{t}\right)^{*}\omega=\omega.
$$
Taking as $ f $ pushed-up from the same function on $ \widetilde L $ functions on $ Y $
and $ T^{*}\widetilde L $ we conclude that the corresponding vector fields
coincide, hence their flows also coincide. Hence the set of points
on $ Y=T^{*}\widetilde L $ where two symplectic forms coincide is stable respective
to these flows along the foliation. Therefore they coincide
everywhere. In fact we have defined an identification of $ T^{*}B $ and $ Y $
basing on a lagrangian transversal to $ {\Cal F} $.
If $ \widetilde L_{1} $, $ \widetilde L_{2} $ are two transversal lagrangian submanifolds, then
we get two identifications of $ T^{*}B $ and $ Y $, hence an automorphism of
$ \left(Y,\omega\right) $ sending $ \widetilde L_{1} $ onto $ \widetilde L_{2} $. Inversely, any automorphism of $ \left(Y,\omega\right) $
sends a lagrangian submanifold to a lagrangian submanifold.
Therefore to prove the last part of the theorem it
suffices to prove that the graph of an $ 1 $-form $ \alpha $ (i.e., the submanifold
$ \left\{\left(b,\alpha|_{b}\right)\right\}\subset T^{*}B $) is a lagrangian submanifold iff $ \alpha $ is closed. However,
it is a special case of a more general formula\footnote{In what follows we find several more formulae with similar sense.
Such formulae can be, therefore, a part of some general formalism.}
$$
\Phi_{\alpha}^{*}\omega=d\alpha,\qquad \Phi_{\alpha}\colon B\to T^{*}B\colon b\mapsto\alpha|_{b}\in T_{b}^{*}B.
$$
\qed\enddemo
\remark{Remark } This theorem can easily be restated for the case of a
Poisson manifold. For this we should change
a lagrangian foliation to a foliation whose leaves are contained in
the symplectic leaves and whose restriction on any symplectic leaf
is a lagrangian foliation. A lagrangian transversal submanifold
changes to an {\it coisotropic\/} transversal, i.e., a submanifold $ \widetilde L $ such that
the form $ \eta $ is 0 on any two conormal to $ \widetilde L $ covectors. In what follows we will
freely use this generalization. \endremark
\head Two maps \endhead
\subhead Reduction of a bihamiltonian structure \endsubhead Let us consider a
bihamiltonian structure on a manifold $ Y $. In each cotangent
space $ T_{y}^{*}Y $ a pair of skew-symmetrical bilinear forms is defined.
Let us call a point $ y\in Y $ a regular point, if the values of Poisson
structures in this point $ \eta_{1}|_{y},\eta_{2}|_{y}\in\Lambda^{2}T_{y}Y $ are in general position as
a pair of bilinear skewsymmetrical forms on $ T_{y}^{*}Y $. The
regular points form an open subset of $ Y $.
From now on we suppose that $ \dim Y $ is odd, $ \dim Y=2k+1 $. We will
consider the subset of regular points only\footnote{Note, that in general case this subset can be empty.}, so we suppose that {\it all\/}
$ Y $ consists of regular points only. Call such a bihamiltonian
structure {\it a structure in general position}. In this case the theorem on
linear algebra shows that in any cotangent space $ T_{y}^{*}Y $ there is a
canonically defined subspace $ W_{y} $.
This
subspace is
generated by $ \operatorname{Ker}\left(l_{1}\eta_{1}+l_{2}\eta_{2}\right) $, hence its orthogonal complement $ W_{y}^{\perp} $ is
the intersection of different $ \operatorname{Ker}\left(l_{1}\eta_{1}+l_{2}\eta_{2}\right)^{\perp} $. But the last spaces
are tangent spaces to the foliations associated with Poisson
structures $ l_{1}\eta_{1}+l_{2}\eta_{2}=\eta_{\lambda} $, $ \lambda=\left(l_{1},l_{2}\right) $. Moreover, since one-dimensional
subspaces $ \operatorname{Ker}\left(l_{1}\eta_{1}+l_{2}\eta_{2}\right) $ form a Veroneze curve, any $ k+1 $ of them
corresponding to $ k+1 $ different values $ \lambda_{1},\lambda_{2},\dots ,\lambda_{k+1}\in P{\Cal S} $ of $ \left(l_{1}:l_{2}\right) $
form a basis in $ W_{x} $. Therefore the corresponding foliations
$ {\Cal F}_{\lambda_{1}},{\Cal F}_{\lambda_{2}},\dots ,{\Cal F}_{\lambda_{k+1}} $ intersect themselves transversally and the tangent
space to this intersection $ \overline{\Cal F} $ at $ y $ is $ W_{y}^{\perp} $. We come to the following
\proclaim{ Theorem } The subspaces $ W_{y}\subset T_{y}^{*}Y $ form orthogonal complements to
a foliation $ \overline{\Cal F} $. On the base of this foliation a Veroneze web is
canonically defined. \endproclaim
\demo{Proof } The first part is already established. To prove the second it
suffices to note that $ \overline{\Cal F} $ being a sub-foliation of any
foliation $ {\Cal F}_{\lambda} $, $ {\Cal F}_{\lambda} $ can be pulled down to a foliation on the
(local) base $ X $ of the foliation $ \overline{\Cal F} $. The resulting foliation on $ X $ will be
denoted by the same letter---this must not lead to a
misunderstanding.
If we fix a point $ y\in Y $ then for its image $ x=\pi\left(y\right) $ on $ X $ (here $ \pi $
being the projection $ Y\to ${\it X\/}) the space $ T_{x}^{*}X $ is canonically isomorphic to
$ W_{y} $. Hence
the orthogonal complements to the leaves of foliations $ {\Cal F}_{\lambda} $ on $ X $
correspond by this isomorphism to $ \operatorname{Ker} \eta_{\lambda} $. Therefore they form a
Veroneze curve. \qed\enddemo
\definition{Definition} This locally defined Veroneze web $ \left(X,{\Cal F}_{\lambda}\right) $ is called
{\it the reduction of bihamiltonian manifold\/} $ \left(Y,\eta_{\lambda}\right) $, $ \left(X,{\Cal F}_{\lambda}\right)={\goth R}_{Y}={\goth R}_{\left(Y,\eta_{\lambda}\right)} $. \enddefinition
\subhead A reverse map \endsubhead Here we will show how to construct a bihamiltonian
structure basing on a Veroneze web $ X $.
Recall, that $ \operatorname{SL}\left({\Cal S}\right) $ denotes a special linear group of the space
$ {\Cal S} $.
\proclaim{ Lemma } On any tangent to $ X $ space $ T_{x}X $ the action of $ \operatorname{SL}\left({\Cal S}\right) $ is
canonically defined. \endproclaim
\demo{Proof } This is already discussed in the section on Veroneze
curves. \qed\enddemo
Now we are going to introduce a notion of {\it associated bundles\/} to the
tangent bundle to $ X $. Begin with a definition of an operation over
vector spaces.
Let $ V $ be an irreducible $ \operatorname{SL}\left({\Cal S}\right) $-module with the highest
weight $ k $ and $ l\in{\Bbb Z} $, $ l\geq-k $. Let $ V^{\left(l\right)} $ denote the only $ \operatorname{SL}\left({\Cal S}\right) $-irreducible
component of $ V\otimes S^{|l|}{\Cal S} $ with the highest weight $ k+l $. It is clear that
$ V^{\left(0\right)}\simeq V $ canonically and that there exists a
canonical morphism $ i_{l}\colon V\otimes S^{|l|}{\Cal S}\to V^{\left(l\right)} $.
\definition{Definition} Denote a vector bundle formed by the vector spaces
$ T_{x}^{*\left(l\right)}X\relSS{=}^{\text{def}}\left(T_{x}^{*}X\right)^{\left(l\right)} $ as $ T^{*\left(l\right)}X $. \enddefinition
In fact we have considered the bundle $ T^{*}X $ as a $ \operatorname{SL}\left({\Cal S}\right) $-bundle and
realized a {\it change of a fiber\/} using the described above explicit
construction\footnote{This explicit construction make it possible to define geometric
objects on the bundle with a changed fiber.}.
We will define the
underlying manifold of the bihamiltonian
structure corresponding to a Veroneze web $ \left(X,{\Cal F}_{\lambda}\right) $ as follows.
\definition{Definition} Let $ Y_{X}= T^{*\left(-1\right)}X $ as a manifold. (Two Poisson structures
on $ Y $ will be defined later.) \enddefinition
To define the Poisson structures on $ Y_{X} $, let us first note that
since a Poisson structure on the cotangent bundle {\it is\/} canonically
defined, on a cotangent bundle $ T^{*}{\Cal F} $ for a foliation $ {\Cal F} $ (which is
a union of cotangent spaces to foliation leaves) a Poisson
structure is also defined as on a union of Poisson manifolds (see
the section on Poisson structures). Therefore a Poisson structure
$ \widetilde\eta_{\lambda} $ is defined on $ T^{*}{\Cal F}_{\lambda} $. It remains only to establish a connection
between $ T^{*}{\Cal F}_{\lambda} $ and $ T^{*\left(-1\right)}X $.
Let $ V $ be an $ \operatorname{SL}\left({\Cal S}\right) $-module, $ 0\not=\lambda\in{\Cal S} $ and $ \widetilde\lambda $ be the corresponding point
in $ P{\Cal S} $, $ B=\operatorname{Stab} \widetilde\lambda $ be a Borel subgroup in $ \operatorname{SL}\left({\Cal S}\right) $ and $ w\in V $ is a
$ B $-highest-weight vector. Then the map
$$
V\to V^{\left(-1\right)}\colon v\mapsto i_{-1}\left(v\otimes\lambda\right),\quad \text{where }i_{l}\colon V\otimes S^{|l|}{\Cal S}\to V^{\left(l\right)},
$$
sends $ V/w $ to $ V^{\left(-1\right)} $ isomorphically. Therefore for every $ \lambda $ the space
$ T_{x}^{*}{\Cal F}_{\lambda} $ is mapped to the space $ T_{x}^{*\left(-1\right)}X $ as a factor of $ T_{x}^{*}X $ by the
space spanned by a highest weight vector.
This identification gives us a Poisson structure
on $ T^{*\left(-1\right)}X $ for any fixed $ \lambda\in V $. Since this identification is of homogeneity
degree 1 in $ \lambda $,
the corresponding structure $ \eta_{\lambda} $ is of homogeneity degree 1 in $ \lambda $.
Therefore, if $ \alpha_{1} $, $ \alpha_{2} $ are fixed, then $ \eta_{\lambda}\left(\alpha_{1},\alpha_{2}\right) $ is a function on $ {\Cal S}\smallsetminus\left\{0\right\} $
of homogeneity degree 1. Being algebraic, $ \eta_{\lambda} $ is
a linear function in $ \lambda $. So we get a bihamiltonian structure on $ Y_{X}=T^{*\left(-1\right)}X $.
\definition{Definition} Let $ e_{1},e_{2} $ be a basis of $ {\Cal S} $. Then a {\it corresponding to\/} $ X $
{\it bihamiltonian structure\/} $ Y_{X} $ is defined as $ \left(T^{*\left(-1\right)}X,\eta_{e_{1}},\eta_{e_{2}}\right) $. \enddefinition
\remark{Remark } It is easy to establish $ \eta_{\lambda} $ being
algebraic in $ \lambda $, since
it depends only on the $ m $-jet of Veroneze web, where $ m $ is finite. Since
the theorem on linear algebra allows the field extension, the
above construction can be applied over the algebraic closure of the
field in question. Therefore this algebraic function has no
singularity. \endremark
Thus, to every Veroneze web on a manifold $ X $ of dimension $ k+1 $ we
associate a bihamiltonian structure on manifold $ Y_{X} $ of dimension
$ 2k+1 $. It is clear that the foliation on $ Y_{X} $ associated with the Poisson
bracket $ \eta_{\lambda} $ is the pull-up of the foliation $ {\Cal F}_{\lambda} $ on $ X $. Therefore the
reduction of this bihamiltonian structure is the original Veroneze
web on $ X $. Hence we have constructed a right inverse map to the
reduction map
$ Y\to{\goth R}_{Y} $. Our next task is to show that locally this map is
completely inverse to reduction, i.e., every bihamiltonian
manifold of odd dimension in general (in the specified above sense)
position can be (locally) obtained basing on this construction.
\remark{Remark } This being a key definition of this article, we
wanted to give a clearer construction. However, we couldn't. So this
definition remains mystique. If a reader isn't satisfied, he can
try to construct these bivector field in a local frame. We couldn't
do this since it is difficult to understand which local frame is
appropriate for consideration of a Veroneze web.
We can
easily define this two bivector fields in points on the zero section\footnote{Really, the tangent space to a linear bundle at such a point is a
canonical direct sum of a tangent space to the base and of the fiber
space. We
can denote the tangent space to the base as $ V $. Then the tangent
space to the bundle is $ W=V\oplus\left(V^{*}\right)^{\left(-1\right)}=V\oplus\left(V^{\left(-1\right)}\right)^{*} $.
\endgraf
Hence $ \Lambda^{2}W=\Lambda^{2}V\oplus V\otimes\left(V^{\left(-1\right)}\right)^{*}\oplus\Lambda^{2}\left(V^{*}\right)^{\left(-1\right)} $. The both bivectors lie in
the middle addend, hence they correspond to maps $ V\to V^{\left(-1\right)} $. If we
consider this pair of bivectors as an element of $ {\Cal S}\otimes\Lambda^{2}V $, then it
corresponds to a natural projection map $ {\Cal S}\otimes V\to V^{\left(-1\right)} $.}. To
define them in the points
outside this section we could consider the translations along the
fibers of this bundle that preserve the bihamiltonian structure we
want to define\footnote{For example, it is one of the simplest possible ways to define a
Poisson (or symplectic) structure on a cotangent bundle $ TZ. $ In this
case the allowed translations ``correspond'' to closed $ 1 $-forms on $ Z $---%
see section on the Poisson geometry.}. However, although we know that there is
sufficiently many such translations, we cannot justify if {\it right now}. \endremark
\head{} Double complex \endhead
\subhead The bicomplex \endsubhead The identification of $ T^{*\left(-1\right)}X $
with $ T^{*}{\Cal F}_{\lambda} $ enables
us to construct a remarkable complex of differential operators on $ X $.
To do this let us consider $ \lambda\in{\Cal S} $ and the foliation $ {\Cal F}_{\lambda} $ on $ X $. The
identification
$$
T^{*\left(-1\right)}X \simeq T^{*}{\Cal F}_{\lambda}
$$
of homogeneity degree $ -1 $ in $ \lambda $ gives rise to an identification
$$
\Lambda^{m}T^{*\left(-1\right)}X \simeq \Omega^{m}{\Cal F}_{\lambda}
$$
of homogeneity degree $ -m $ in $ \lambda $. (Here $ \Omega^{m}{\Cal F}_{\lambda} $ denotes
$ \Lambda^{m}T^{*}{\Cal F}_{\lambda} $.) Therefore the operator $ d $ of exterior differentiation
induces a first order differential operator
$$
d_{\lambda}\colon \Gamma\left(\Lambda^{m}T^{*\left(-1\right)}X\right) \to \Gamma\left(\Lambda^{m+1}T^{*\left(-1\right)}X\right)
$$
of homogeneity degree 1 in $ \lambda $. Hence the above consideration
shows that this operator must be linear in $ \lambda $. Since $ d_{\lambda}^{2}\equiv 0 $,
$$
d_{\lambda}d_{\mu}+d_{\mu}d_{\lambda}=\left(d_{\lambda}+d_{\mu}\right)^{2}-d_{\lambda}^{2}-d_{\mu}^{2}=0
$$
for any $ \lambda,\mu\in{\Cal S} $. Therefore for fixed $ \lambda $ and $ \mu $ we get a bicomplex
$$
\CD \dots @. \dots @. \dots
\\
\SubSup!u{d_{\mu}} \SubSup!u{d_{\mu}} \SubSup!u{d_{\mu}}
\\
\Lambda^{2}T^{*\left(-1\right)}X \SubSup!R{d_{\lambda}} \Lambda^{3}T^{*\left(-1\right)}X \SubSup!R{d_{\lambda}} \Lambda^{4}T^{*\left(-1\right)}X \SubSup!R{d_{\lambda}} \dots
\\
\SubSup!u{d_{\mu}} \SubSup!u{d_{\mu}} \SubSup!u{d_{\mu}}
\\
\Lambda^{1}T^{*\left(-1\right)}X \SubSup!R{d_{\lambda}} \Lambda^{2}T^{*\left(-1\right)}X \SubSup!R{d_{\lambda}} \Lambda^{3}T^{*\left(-1\right)}X \SubSup!R{d_{\lambda}} \dots
\\
\SubSup!u{d_{\mu}} \SubSup!u{d_{\mu}} \SubSup!u{d_{\mu}}
\\
\Lambda^{0}T^{*\left(-1\right)}X \SubSup!R{d_{\lambda}} \Lambda^{1}T^{*\left(-1\right)}X \SubSup!R{d_{\lambda}} \Lambda^{2}T^{*\left(-1\right)}X \SubSup!R{d_{\lambda}} \dots \endCD.
$$
This bicomplex will play a crucial role in what follows.
\subhead Objects of differential geometry \endsubhead A notion of
{\it an object of differential geometry\/} lingered for a long time and was
introduced (more or less explicitly) in the works \cite{5}.
\definition{Definition} Let $ {\goth D} $ be a $ {\Bbb Z} $-graded superalgebra\footnote{It means that the even and odd parts of $ {\goth D} $ are the sum of components
with even and odd grading:
$$
{\goth D}^{+}=\sum_{i}{\goth D}_{2i},\quad {\goth D}^{-}=\sum_{i}{\goth D}_{2i+1}.
$$
} with only three
graded components: $ {\goth D}_{-1} $, $ {\goth D}_{0} $ and $ {\goth D}_{1} $. {\it An object of differential
geometry\/} is a $ {\Bbb Z} $-graded representation of $ {\goth D} $. \enddefinition
\example{ Example } The crucial example of an object of differential
geometry is the de Rham complex $ \Omega^{*} $ for a manifold $ Z $. Here $ {\goth D}_{1} $ is
generated by a single element $ d $, which acts as the exterior
differentiation on $ \Omega^{*} $. Both spaces $ {\goth D}_{-1} $ and $ {\goth D}_{0} $ are identified with
the space $ \operatorname{Vect}\left(Z\right) $ of global vector fields on $ Z $. $ {\goth D}_{-1} $ acts on $ \Omega^{*} $ by
the inner multiplication and $ {\goth D}_{0} $ acts by Lie differentiation. The
Leibnitz identity
$$
{\Cal L}_{v}=i_{v}\circ d+ d\circ i_{v},
$$
where $ {\Cal L}_{v} $ is Lie differentiation and $ i_{v} $ is the inner multiplication
together with
$$
i_{v_{1}}i_{v_{2}}+i_{v_{2}}i_{v_{1}}=0,\quad d^{2}=0
$$
shows that $ {\goth D} $ with the structure of Lie superalgebra inherited from
action on $ \Omega^{*} $ does satisfy the conditions of definition. \endexample
The notion of an object of differential geometry is an
alternative to the description of geometry based on the ring
structure on the space of (global) functions. The latter
structure seems exceeding in the case, say, of the variational
calculus where the functions in question (i.e., integrals of local
functionals) don't allow the multiplication.
Though the spaces $ {\goth D}_{-1} $ and $ {\goth D}_{1} $ are completely symmetrical
in the
above definition, they are very different in the example: $ {\goth D}_{-1} $
is infinite-dimensional and the $ {\goth D}_{1} $ is one-dimensional. It
is easy to see that the procedure described in the previous section
gives an
example of an object of differential geometry with {\it the
two-dimensional\/}
space $ {\goth D}_{1} $. Indeed, let $ {\goth D}_{-1} $ be the space of $ T^{\left(-1\right)}X=\left(T^{*\left(-1\right)}X\right)^{*} $ sections
and let $ {\goth D}_{1} $ be generated by $ d_{\lambda} $ and $ d_{\mu} $. Then they are both acting on
$ \Lambda^{*}T^{*\left(-1\right)}X $:
the first one by inner multiplication, the second one by
differentiation. Since
$$
i_{v_{1}}i_{v_{2}}+i_{v_{2}}i_{v_{1}}=0,\quad d_{\lambda}^{2}=d_{\mu}^{2}=d_{\lambda}d_{\mu}+d_{\mu}d_{\lambda}=0,
$$
these spaces of operators in $ \Lambda^{*}T^{*\left(-1\right)}X
$ generate together a $ {\Bbb Z} $-graded superalgebra with
components in the gradings $ -1 $, 0 and 1 only. Thus
the structure of a differential geometry object is defined on $ \Lambda^{*}T^{*\left(-1\right)}X $.
\subhead Double cohomology \endsubhead \definition{Definition} A double complex is a graded
linear space $ \oplus A^{i} $, $ i\in{\Bbb Z} $, and a differential $ d_{\lambda} $ on $ A^{*} $ that depends linearly on
a vector $ \lambda\in{\Cal S} $. Here $ {\Cal S} $ is a fixed two-dimensional space. \enddefinition
\remark{Remark } The identity $ d_{\lambda}^{2}=0 $ implies that
$$
d_{\mu}d_{\lambda}+d_{\lambda}d_{\mu}=0.
$$
\endremark
We define double cohomology of a double complex $ \left(A^{*},d_{\lambda}\right) $ as
$$
{\Bbb H}^{i}\left(A\right)=\frac{\operatorname{Ker} d_{\nu_{1}}d_{\nu_{2}}\colon A^{i}\to A^{i+2}\otimes\Lambda^{2}{\Cal S}^{*}}{\operatorname{Im} d_{\nu}\colon A^{i-1}\otimes{\Cal S}\to A^{i}}.
$$
\remark{Remark } There is another related definition of double cohomology:
$$
\widetilde{\Bbb H}^{i}\left(A\right)=\frac{\operatorname{Ker} d_{\nu}\colon A^{i}\to A^{i+1}\otimes{\Cal S}^{*}}{\operatorname{Im} d_{\nu_{1}}d_{\nu_{2}}\colon A^{i-2}\otimes\Lambda^{2}{\Cal S}\to A^{i}}.
$$
The definition we have chosen seems a little more useful.
Clearly for fixed $ \lambda\in{\Cal S} $ there is a natural map $ d_{\lambda}:
{\Bbb H}^{i-1}\left(A\right)\to\widetilde{\Bbb H}^{i}\left(A\right) $. If $ \left(A^{*},d_{\lambda}\right) $
for every $ \lambda\not=0 $ is exact in the degree $ i $, then this map is isomorphism for
this $ i $.
Really, if $ d_{\lambda}a^{i}=0 $, $ \lambda\in{\Cal S} $, then for any $ 0\not=\nu\in{\Cal S} $ there exists $ b_{\nu}^{i-1} $
such that
$ d_{\nu}b_{\nu}^{i-1}=a^{i} $. Since $ d_{\lambda}d_{\mu}b_{\nu}^{i-1}=0 $ for $ \lambda,\mu,\nu\in{\Cal S} $, the considered map is
surjective. The injectivity of this map is obvious. \endremark
\head Twist of a bihamiltonian structure \endhead We have built a bihamiltonian
structure associated to a Veroneze web on a manifold. In fact, there
exists a whole family of such structures, all of them having the same
reduction. To construct a member of this family it will be rather
useful to
introduce the notion of {\it a twisted bihamiltonian structure}.
\definition{Definition} Let $ \left(Y,\eta_{\lambda}\right) $ be a bihamiltonian manifold. Let $ X={\goth R}_{Y} $ be its
reduction and let $ f $ be a function on $ X $ considered as a section
of $ \Lambda^{0}T^{*\left(-1\right)}X $. Suppose $ f $ be a ``double cocycle'' in the complex associated
with the above bicomplex, i.e., $ d_{\lambda}d_{\mu}f=0 $. Let us call the
bihamiltonian manifold $ \left(Y,\lambda_{1}\eta_{1}+\lambda_{2}{\goth T}_{d_{\mu}f*}\eta_{2}\right) $ {\it the twist of\/} $ Y $ {\it with the
cocycle\/} $ f $. Here $ \eta_{1} $, $ \eta_{2} $ are two Poisson structures associated with
the basis of $ {\Cal S} $, $ \lambda_{1} $, $ \lambda_{2} $ are the corresponding coordinate functions in this
basis, $ {\goth T}_{\phi} $
for a section $ \phi $ of a linear bundle $ {\Cal B} $ denotes the translation on this section
in the bundle:
$$
{\goth T}_{\phi}\left(\left(x,v\right)\right)=\left(x,v+\phi\left(x\right)\right),\qquad v\in{\Cal B}_{x}.
$$
\enddefinition
\remark{Remark } To prove that both bivectors fields form Poisson
structures it is sufficient to note that the direct image of
a Poisson structure is a Poisson structure again. However, to prove the
agreement of these structures between themselves is much more
difficult. The proof will be given in the next section, after
an extensive study of the geometry of a bihamiltonian manifold. \endremark
\head Geometry of a bihamiltonian structure \endhead First, we will study
geometry of a leaf of the foliation we have defined on a
bihamiltonian manifold.
\subhead An affine structure \endsubhead The first target is to prove the following
\proclaim{ Theorem } On each leaf of foliation $ \widetilde{\Cal F} $ on bihamiltonian manifold $ Y $
a canonical affine structure can be defined\footnote{It is this structure that allows considering such a manifold as an
integrable system.}. \endproclaim
\demo{Proof } Since $ \eta_{\lambda} $ maps the conormal space $ N_{y}^{*}\widetilde{\Cal F}_{y} $ for the leaf $ \widetilde{\Cal F}_{y} $
of foliation $ \widetilde{\Cal F} $ which contains point $ y\in Y $ into the tangent space
$ T_{y}\widetilde{\Cal F}_{y} $ for this leaf:
$$
\eta_{\lambda}\colon N_{y}^{*}\widetilde{\Cal F}_{y}\to T_{y}\widetilde{\Cal F}_{y}
$$
(compare with the theorem on linear algebra) and since $ N_{y}^{*}\widetilde{\Cal F}_{y} $ coincides with
$ T_{x}^{*}X $ (where $ x=\pi\left(y\right){} $, $ \pi\colon Y\to{\goth R}_{Y}=X $), the
tangent spaces to $ \widetilde{\Cal F}_{y} $ at different points are all identified with
$ T_{x}^{*}X/\operatorname{Ker}\left(\eta_{\lambda}\right) $, where $ \eta_{\lambda} $ is considered as a map $ T_{x}^{*}X\to T_{y}\widetilde{\Cal F}_{y} $. Since
this kernel is just $ N_{x}^{*}{\Cal F}_{\lambda} $ (here $ {\Cal F}_{\lambda} $ is considered earlier foliation on
$ X={\goth R}_{Y} $) and doesn't depend on the point on $ \widetilde{\Cal F}_{y}=\pi^{-1}\left(x\right) $, we obtain a
canonical identification of these tangent spaces at different points
of $ \widetilde{\Cal F}_{y} $. Thus a flat affine connection on $ \widetilde{\Cal F}_{y} $ is canonically
defined. Hence the only thing to prove now is to show that the
torsion of this connection equals to 0.
Let $ f $, $ g $ be two functions on $ X $. Then
$$
\left\{\pi^{*}\left(f\right),\pi^{*}\left(g\right)\right\}_{\lambda}=0
$$
since $ d\left(\pi^{*}\left(f\right)\right) $ and $ d\left(\pi^{*}\left(g\right)\right) $ both lie in the space $ W $ from the theorem
on linear algebra, which is isotropic relative to any bilinear
form. Therefore the vector fields $ L_{\pi}^{\lambda}*_{\left(f\right)} $, $ L_{\pi}^{\lambda}*_{\left(g\right)} $ (where
$$
L_{\varphi}^{\lambda}\cdot\psi=\left\{\varphi,\psi\right\}_{\lambda},\quad \varphi\text{ and }\psi\text{ being functions on {\it Y\/})}
$$
commute. By the above consideration these
vector fields are constant along $ \widetilde{\Cal F}_{y} $ relative to the considered
connection, hence this commutator is the value of the torsion on
these
vectors. Since the vectors of the specified type fill the whole tangent
space to $ \widetilde{\Cal F}_{y} $, the torsion is identically 0, what proves the
theorem, if we show that the defined structure doesn't depend on $ \lambda\in{\Cal S} $.
Anyway, for fixed $ \lambda $
on the reduction $ X={\goth R}_{Y} $ is defined
a canonical bundle with an affine fiber. Let us consider the
associated vector bundle $ {\goth L} $.
\proclaim{ Lemma } This vector bundle is canonically isomorphic to $ T^{*\left(-1\right)}X $.
This isomorphism doesn't depend on $ \lambda $. \endproclaim
\demo{Proof } Fix a point $ y\in Y $, let $ x=\pi\left(y\right) $. Then $ \eta_{\lambda} $ determines a map
$ T_{x}^{*}X\otimes{\Cal S}\to T_{y}\widetilde{\Cal F} $. For any $ \lambda\in{\Cal S} $ the subspace $ \operatorname{Ker} \left(\eta_{\lambda}\right)\otimes\lambda $ lies in the kernel
of this map. Moreover, the kernel is generated by these
subspaces. Consider the spaces $ T_{x}^{*}X $, $ {\Cal S} $ and $ T_{y}\widetilde{\Cal F} $ as $ \operatorname{SL}\left({\Cal S}\right) $-modules.
The structures of a module on $ T_{x}^{*}X $ and on $ T_{y}\widetilde{\Cal F} $ are defined by the
Veroneze curves in these spaces: $ \lambda\mapsto\operatorname{Ker} \left(\eta_{\lambda}\right) $ in the first and
$ \lambda\mapsto\operatorname{Im}\left(\operatorname{Ker} \left(\eta_{\lambda}\right)\otimes{\Cal S}\right) $ in the second. A remark after the theorem on linear
algebra shows that the map $ T_{x}^{*}X\otimes{\Cal S}\to T_{y}\widetilde{\Cal F} $ is $ \operatorname{SL}\left({\Cal S}\right) $-covariant. Since
the domain of this map is a sum of two irreducible components, this
map is essentially the projection on one of this components $ -
\left(T_{x}^{*}X\right)^{\left(-1\right)} $,
i.e., there is an identification
$$
\left(T_{x}^{*}X\right)^{\left(-1\right)}\simeq T_{y}{\Cal F}.
$$
The above discussion shows that this identification taken at
different points of $ \pi^{-1}\left(x\right) $ leads to the same affine structure on
$ \pi^{-1}\left(x\right) $. Hence there is an identification $ \left(T_{x}^{*}X\right)^{\left(-1\right)}\to{\goth L} $. \qed\enddemo
This shows that affine structure on leaves really doesn't
depend on $ \lambda\in{\Cal S} $. \qed\enddemo
This identification transfers the bihamiltonian structure on
$ \left(T^{*}X\right)^{\left(-1\right)} $ to a
bihamiltonian structure on $ {\goth L} $.
We can apply this construction to the example of
a bihamiltonian structure---to the total space of the bundle
$ T^{*\left(-1\right)}X $. It is easy to see that in this case we get nothing new\footnote{Let us recall that $ {\goth R}_{T^{*\left(-1\right)}X}=X $.}:
\proclaim{ Lemma } The considered above affine structure on the fibers of the
bundle $ T^{*\left(-1\right)}X $ is associated with the linear structure on this
bundle. \endproclaim
In fact now we have shown
\roster
\item
how to construct a new bihamiltonian
manifold $ T^{*\left(-1\right)}{\goth R}_{Y} $ basing on a bihamiltonian manifold $ Y $;
\item
that this operation is an idempotent operation;
\item
that the old manifold is connected with the new as an bundle
with an affine fiber is connected with the associated vector bundle.
\endroster
In fact we want to show that these bihamiltonian manifolds are
isomorphic. Both this manifold are affine foliations over the same
base. To identify them it is sufficient to choose a section of $ Y\to X $
that corresponds to the zero section of $ T^{*\left(-1\right)}X\to X $.
\subhead A connection with lagrangian foliations \endsubhead Let us first fix $ \lambda\in{\Cal S} $
and consider {\it one\/} Poisson
structure $ \eta_{\lambda} $. Any leaf of the corresponding foliation $ {\Cal F}_{\lambda} $ is
fibered (by $ \widetilde{\Cal F} $) over $ X $
(with the corresponding foliation denoted by the same symbol $ {\Cal F}_{\lambda} $).
Since any two functions on $ X $ are in involution and the required
restraints on dimensions are satisfied, the fibers of this foliation
are lagrangian inside leaves of the first foliation. It is known
that Hence these (lagrangian) fibers are equipped with a canonical affine
structure.
Now we are going to show that in fact these two defined on
leaves of $ \widetilde{\Cal F} $ affine structures coincide. To do this, it is
sufficient to note that the first identification connects $ T_{y}\widetilde{\Cal F}_{y} $ with
$ T_{x}^{*}X/\operatorname{Ker}\left(\eta_{\lambda}\right) $, where $ \eta_{\lambda} $ is considered as a map $ T_{x}^{*}X\to T_{y}\widetilde{\Cal F}_{y} $, $ y=\pi\left(x\right) $.
The second one connects this space with $ T_{x}^{*}{\Cal F}_{\lambda,x} $, (here $ {\Cal F}_{\lambda,x} $ is the
containing the point $ x $ leaf of the foliation $ {\Cal F}_{\lambda} $ on {\it X\/}). These two
spaces are obviously canonically isomorphic and this isomorphism
makes the diagram
$$
\CD T_{y}\widetilde{\Cal F}_{y} @>>> T_{x}^{*}X/\operatorname{Ker}\left(\eta_{\lambda}\right)
\\
@| @VVV
\\
T_{y}\widetilde{\Cal F}_{y} @>>> T_{x}^{*}{\Cal F}_{\lambda,x} \endCD
$$
commutative, which proves the assertion.
\proclaim{ Corollary } Let us fix two points $ \lambda,\mu\in{\Cal S} $. Then on a leaf of the
foliation $ \widetilde{\Cal F} $ we can consider two affine structures, associated with
structures of lagrangian foliations on $ {\Cal F} $ respective to Poisson
structures $ \eta_{\lambda} $ and $ \eta_{\mu} $. These two affine structures coincide. \endproclaim
\remark{Remark } We have proved this corollary by describing this affine
structure in independent of $ \lambda $ terms. However, we cannot prove this
miraculous fact more directly. \endremark
\subhead Translations and Schouten---Nijenhuis bracket \endsubhead Darboux
theorem (on the straightening of a symplectic
structure) shows that there exist a lagrangian submanifold
transversal to
given foliation.
Similar
arguments make evident the existence of a transversal for a Poisson manifold
in general position (which is foliated on symplectic manifolds).
Let us consider two Poisson structures $ \eta_{\lambda} $, $ \eta_{\mu} $, $ \lambda,\mu\in{\Cal S} $. The above
arguments show that we can find two transversal to a foliation $ {\Cal F} $
submanifold, one coisotropic respective to $ \eta_{\lambda} $, another respective to
$ \eta_{\mu} $. The theorem on Poisson structures,
applied to the first of them, identify $ \left(Y,\eta_{\lambda}\right) $ with $ \left(T^{*\left(-1\right)}X,\eta_{\lambda}\right) $. The
second transversal corresponds
to a section $ t $ of $ T^{*\left(-1\right)}X $ under this identification. Hence,
$$
\left(Y,\eta_{\lambda},\eta_{\mu}\right) \simeq \left(T^{*\left(-1\right)}X,\eta_{\lambda},{\goth T}_{t*}\eta_{\mu}\right),
$$
where $ {\goth T}_{t} $ is the shift on the section $ t $ on the vector bundle $ T^{*\left(-1\right)}X $:
$$
{\goth T}_{t}\colon \left(x,\alpha\right)\mapsto\left(x,a+t\left(x\right)\right).
$$
The next problem to consider is the following
\proclaim{ Theorem } For a Veroneze web $ X $ and sections $ t_{1} $, $ t_{2} $ of $ T^{*\left(-1\right)}X $ the
manifold
$$
\left(T^{*\left(-1\right)}X,{\goth T}_{t_{1}*}\eta_{\lambda},{\goth T}_{t_{2}*}\eta_{\mu}\right)
$$
with two Poisson structures is bihamiltonian iff the sections $ t_{1,2} $
satisfy the equation
$$
d_{\lambda}d_{\mu}\left(t_{1}-t_{2}\right)=0,
$$
here $ d_{\lambda} $ and $ d_{\mu} $ are two differential in the considered above
bicomplex.
In this case the bihamiltonian structure coincides (after a
diffeomorphism of $ T^{*\left(-1\right)}X $) with the standard bihamiltonian structure
on $ T^{*\left(-1\right)}X $ twisted with $ t_{2}-t_{1} $:
$$
\left(T^{*\left(-1\right)}X,{\goth T}_{t_{1}*}\eta_{\lambda},{\goth T}_{t_{2}*}\eta_{\mu}\right)\simeq{\goth T}_{t_{1}}\left(T^{*\left(-1\right)}X,\eta_{\lambda},{\goth T}_{\left(t_{2}-t_{1}\right)*}\eta_{\mu}\right).
$$
For $ t_{1}-t_{2}=-d_{\lambda}f_{1}+d_{\mu}f_{2} $\footnote{Such $ t $ clearly satisfies the above equation.}, $ f_{1,2} $ being sections of $ \Lambda^{0}T^{\left(-1\right)}X $ (this
space coincides with functions on X), this bihamiltonian structure
coincides (after a diffeomorphism of $ T^{*\left(-1\right)}X $) with the standard
bihamiltonian structure on $ T^{*\left(-1\right)}X $:
$$
\left(T^{*\left(-1\right)}X,{\goth T}_{t_{1}*}\eta_{\lambda},{\goth T}_{t_{2}*}\eta_{\mu}\right)\simeq{\goth T}_{t_{1}+d_{\lambda}f_{1}} \left(T^{*\left(-1\right)}X,\eta_{\lambda},\eta_{\mu}\right).
$$
\endproclaim
\demo{Proof } The last assertion is almost obvious. As it was
proved in the section on Poisson structures, the
translation on the differential of the function in conormal bundle
to a manifold
$$
\left(x,\xi\right)\mapsto\left(x,\xi+df\left(x\right)\right)
$$
preserves the symplectic (hence also the Poisson) structure.
Clearly, the same is true for a conormal bundle to a foliation (with
the change of symplectic and Poisson structures). Hence the
isomorphism of $ \left(T^{*\left(-1\right)}X,\eta_{\lambda}\right) $ and $ \left(T^{*}{\Cal F}_{\lambda},\eta_{\lambda}\right) $ (together with the
corresponding isomorphism with a change $ \lambda $ and $ \mu $) shows that
$$
{\goth T}_{d_{\lambda}f_{1}*}\eta_{\lambda}=\eta_{\lambda},\quad {\goth T}_{d_{\mu}f_{1}*}\eta_{\mu}=\eta_{\mu},
$$
what proves the last formula of the theorem.
Inversely, the translation along the affine bundle clearly
doesn't change the field of $ 2 $-vectors' kernels.
Let $ \Phi $ be a
diffeomorphism such that
$$
\left(T^{*\left(-1\right)}X,{\goth T}_{t_{1}*}\eta_{\lambda},{\goth T}_{t_{2}*}\eta_{\mu}\right)\simeq\Phi\left(T^{*\left(-1\right)}X,\eta_{\lambda},\eta_{\mu}\right).
$$
Since the projection $ T^{*\left(-1\right)}X\to X $ is defined by the bihamiltonian
structure $ \left(T^{*\left(-1\right)}X,\eta_{\lambda},\eta_{\mu}\right) $, $ \Phi $ sends a leaf of the foliation $ \widetilde{\Cal F} $
on $ T^{*\left(-1\right)}X $ to a leaf. Hence $ \Phi $ induces the diffeomorphism $ \Phi_{X} $ of the Veroneze
web on $ X $. We can suppose this diffeomorphism to be identical since
$$
\widetilde{\Phi_{X}}\left(T^{*\left(-1\right)}X,\eta_{\lambda},\eta_{\mu}\right)\simeq\left(T^{*\left(-1\right)}X,\eta_{\lambda},\eta_{\mu}\right)
$$
and $ \Phi $ can be changed to $ \Phi\circ\widetilde{\Phi_{X}}^{-1} $, $ \widetilde{\Phi_{X}} $ being the induced by $ \Phi_{X} $
diffeomorphism of $ \left(T^{*\left(-1\right)}X,\eta_{\lambda},\eta_{\mu}\right) $.
The $ \Phi $-image of the zero section must be a coisotropic
respective to any form $ \eta_{\lambda} $ transversal to this
foliation submanifold. Fixing $ \lambda $, we get an
identification of $ {\goth T}_{t_{1}}^{-1}\Phi\left(\text{0-section}\right) $
and the differential along
the foliation $ {\Cal F}_{\lambda} $
of a function $ f_{1} $:
$$
{\goth T}_{t_{1}}^{-1}\Phi\left(\text{0-section}\right)=\Gamma_{d_{\lambda}f_{1}},
$$
here $ \Gamma_{\varphi} $ being a graph for a section $ \varphi $ of a bundle, $ \Gamma_{\varphi}=\left\{\left(x,\varphi\left(x\right)\right)\right\} $.
The same can
be repeated for a fixed $ \mu $:
$$
{\goth T}_{t_{2}}^{-1}\Phi\left(\text{0-section}\right)=\Gamma_{d_{\mu}f_{2}}.
$$
Hence
$$
t_{1}+d_{\lambda}f_{1}=t_{2}+d_{\mu}f_{2}.
$$
Now it is easy to see that we have
found the functions $ f_{1} $, $ f_{2} $ required in the theorem.
We have considered the last assertion of
the theorem. Let us consider now the first assertion. In fact we
will prove much more stronger assertion:
$$
\left[{\goth T}_{t_{1}}\eta_{\lambda},{\goth T}_{t_{2}}\eta_{\mu}\right]\simeq d_{\lambda}d_{\mu}\left(t_{1}-t_{2}\right),\qquad d_{\lambda}d_{\mu}\colon \Lambda^{1}T^{*\left(-1\right)}X\to\Lambda^{3}T^{*\left(-1\right)}X.
$$
Here the sign $ \simeq $ denotes that the left-hand side is an image of the
tangent to the fibers $ 3 $-vector field in the right-hand side in the
space of $ 3 $-vectors in the total space of the bundle. Application of
$ {\goth T}_{t_{1}}^{-1} $ allows us to to reduce the general case to a case $ t_{1}=0 $, $ t_{2}=t $.
The Schouten---Nijenhuis bracket
$$
\left[\eta_{\lambda},{\goth T}_{t*}\eta_{\mu}\right]
$$
has a decomposition connected with the filtration
$$
0\subset T_{y}\widetilde{\Cal F}\subset T_{y}T^{*\left(-1\right)}X,\qquad T_{y}T^{*\left(-1\right)}X/T_{y}\widetilde{\Cal F}\simeq T_{x}^{*}X,\quad y\in T_{x}^{*\left(-1\right)}X.
$$
The corresponding filtration on $ \Lambda^{3}T_{y}T^{*\left(-1\right)}X $ has associated factors
$$
\Lambda^{3}T_{y}\widetilde{\Cal F},\quad \Lambda^{2}T_{y}\widetilde{\Cal F}\otimes T_{x}^{*}X,\quad T_{y}\widetilde{\Cal F}\otimes\Lambda^{2}T_{x}^{*}X,\quad \Lambda^{3}T_{x}^{*}X.
$$
It is easy to see that {\it for any\/} section $ t $ the components of
$ \left[\eta_{\lambda},{\goth T}_{t*}\eta_{\mu}\right] $ in the last three factors are zeros.
Really, to simplify the expression
$$
2\left[\eta_{\lambda},{\goth T}_{t*}\eta_{\mu}\right] = \operatornamewithlimits{Alt}_{i j m}\eta_{\lambda}^{il}{\goth T}_{t*}\eta_{\mu ,l}^{jm} + \operatornamewithlimits{Alt}_{i j m}{\goth T}_{t*}\eta_{\mu}^{il}\eta_{\lambda}^{jm}{}_{,l},
$$
it is useful to choose a coordinate system $ \left(x_{i},i=1,\dots ,k\right) $ on $ X $ such that the
foliation $ {\Cal F}_{\nu} $ (for an appropriate $ \nu $) is
$ x_{k}=\operatorname{const} $ and that $ \eta $ has constant coefficients. After that we can
choose on $ T^{*\left(-1\right)}X $ the coordinate system
associated with the identification of this bundle and $ T^{*\left(-1\right)}{\Cal F}_{\nu} $.
The $ 2 $-vector $ {\goth T}_{t*}\eta_{\mu}-\eta_{\mu} $ at a fixed point of $ T_{x}^{*\left(-1\right)}X $ is a
tangent to $ \widetilde{\Cal F}\quad 2 $-vector. (It is clear for $ t\left(x\right)=0 $ or $ t=d_{\mu}f $ and
consequently for an arbitrary {\it t}.)
First we want to prove an explicit formula for this difference
$$
{\goth T}_{t*}\eta_{\mu}-\eta_{\mu}=d_{\mu}t,\qquad d_{\mu}\colon \Lambda T^{*\left(-1\right)}X\to\Lambda T^{*\left(-1\right)}X.
$$
Since here we need to consider only one of two Poisson
structures $ \eta_{\mu} $,
we can use the above coordinate system with $ \nu=\mu $ and the
corresponding
identification of $ T^{*\left(-1\right)}X $ and $ T^{*}{\Cal F}_{\mu} $. If $ \phi $ is a section
of the cotangent bundle for a manifold $ Z $ and $ \eta $ is the Poisson structure
on this bundle, then
$$
\left({\goth T}_{\phi}\right)_{*}\eta-\eta=-d\phi,
$$
where $ d\phi $ is considered as a section of $ \Omega^{2}Z=\Lambda^{2}\left(T^{*}Z\right)\subset\Lambda^{2}T\left(T^{*}Z\right) $. Indeed,
it is already proved in the case $ d\phi=0 $, and the proof in the
general case is
completed after considering $ \phi $ such that $ \phi\left(z\right)=0 $. In this case
$$
{\goth T}_{\phi*}|_{\left(z,\zeta\right)}=\left(\matrix \boldkey1 & \boldkey0
\\
\widetilde d\phi & \boldkey1\endmatrix\right),\qquad \left(z,\zeta\right)\in T_{z}^{*}Z
$$
in the coordinate system $ \left(z^{i},\zeta_{i}\right) $ on $ T^{*}Z $. Here $ \left(\widetilde d\phi\right)_{ij}=\phi_{i,j} $. Hence $ \Lambda^{2}{\goth T}_{\phi*} $
has also the lower-diagonal block
structure and it is easy to see that the only important for the
calculation of $ \Lambda^{2}\left({\goth T}_{\phi}\right)_{*}\cdot\eta $ block is
$$
\gather \widetilde d\phi\wedge\boldkey1 =-d\phi\wedge\text{{\bf1},\quad }\left(\widetilde d\phi\right)_{ij}=-\phi_{i,j}+\phi_{j,i},
\\
\left(\widetilde d\phi\wedge\boldkey1\right)\left(\frac{\partial}{\partial z^{j}}\wedge\frac{\partial}{\partial\zeta_{j}}\right) = \left(\widetilde d\phi\right)_{ij}\frac{\partial}{\partial\zeta_{i}}\wedge\frac{\partial}{\partial\zeta_{j}}.\endgather
$$
Since
$$
\left(d\phi\wedge\boldkey1\right)\cdot\eta=d\phi,
$$
the assertion is proved.
The arguments for the case of cotangent bundle to a
foliation are exactly the same. This shows that
$$
\left[\eta_{\lambda},{\goth T}_{t*}\eta_{\mu}\right]=\left[\eta_{\lambda},{\goth T}_{t*}\eta_{\mu}-\eta_{\mu}\right]=-\left[\eta_{\lambda},d_{\mu}t\right]
$$
(we have used the identification of operator $ d $ on $ T^{*}{\Cal F}_{\mu} $ and
operator $ d_{\mu} $ on $ T^{*\left(-1\right)}X $).
In this formula again only one of Poisson structures appears,
hence we can again consider the above identification of $ T^{*\left(-1\right)}X $ and (this
time) $ T^{*}{\Cal F}_{\lambda} $ (i.e., now $ \nu=\lambda $). The section $ d_{\mu}t $ of $ \Lambda^{2}T^{*\left(-1\right)}X $ corresponds
to a $ 2 $-form
$ \omega $ along the foliation $ {\Cal F}_{\lambda} $ (so $ \omega\in\Lambda^{2}T^{*}{\Cal F}_{\lambda} $) and consequently to a tangent
to fibers
$ 2 $-vector field $ \widetilde\omega $ on $ T^{*}{\Cal F}_{\lambda} $, $ \widetilde\omega\left(x,\xi\right)=\omega_{ij}
\frac{\partial}{\partial\zeta_{i}}\wedge\frac{\partial}{\partial\zeta_{j}} $ if $ \omega=\omega_{ij}dx^{i}\wedge dx^{j} $. We claim that on a symplectic
manifold $ Z $
$$
\left[\eta,\widetilde\omega\right]=-\widetilde{d\omega},
$$
where $ \eta $ is the Poisson structure, $ \omega $ is an arbitrary $ 2 $-form on $ Z $,
$ \widetilde\omega $ is
the corresponding $ 2 $-vector field on $ T^{*}Z $, $ \widetilde{d\omega} $ is a
$ 3 $-vector field on $ T^{*}Z $
that corresponds to $ d\omega $ in the same way as $ \widetilde\omega $ corresponds to $ \omega $. The
extension of this formula to the
case of foliation will complete the proof of the theorem.
To prove this formula consider a local coordinate
system on $ Z $. Let us
choose the associated coordinate system
on
$ T^{*}Z $, $ \left(y^{1},\dots ,y^{2k}\right)=\left(z^{1},\dots ,z^{k},\zeta_{1},\dots ,\zeta_{k}\right) $. In this frame the tensor
field $ \eta $ is constant, so
$$
\left[\eta,\widetilde\omega\right]= \operatornamewithlimits{Alt}_{i j m}\eta^{il}\widetilde\omega_{,l}^{jm} + \operatornamewithlimits{Alt}_{i j m}\widetilde\omega^{il}\eta_{,l}^{jm}=\operatornamewithlimits{Alt}_{i j m}\eta^{il}\widetilde\omega_{,l}^{jm}.
$$
The $ 2 $-vector $ \widetilde\omega $ is an image of a tangent to a fiber $ 2 $-vector. Let us
consider a filtration on $ \Lambda^{3}T_{\left(z,\zeta\right)}T^{*}Z $ that is connected with the
filtration
$$
0\subset T_{\left(z,\zeta\right)}{\goth F}\subset T_{\left(z,\zeta\right)}T^{*}Z,\qquad T_{\left(z,\zeta\right)}T^{*}Z/T_{\left(z,\zeta\right)}{\goth F}\simeq T_{z}Z,\quad \left(z,\zeta\right)\in T^{*}Z
$$
on $ T_{\left(z,\zeta\right)}T^{*}Z $ (here $ {\goth F} $ is a vector bundle consisting of tangent to
fibers of projection $ T^{*}Z\to Z $ vectors). It has associated factors
$$
\Lambda^{3}T_{\left(z,\zeta\right)}{\goth F},\quad \Lambda^{2}T_{\left(z,\zeta\right)}{\goth F}\otimes T_{z}Z,\quad T_{\left(z,\zeta\right)}{\goth F}\otimes\Lambda^{2}T_{z}Z,\quad \Lambda^{3}T_{z}Z.
$$
It is easy to see that {\it for any\/} section $ \omega $ of $ \Omega^{2}Z $ the components of
$ \left[\eta,\widetilde\omega\right] $ (where $ \widetilde\omega $ is the corresponding section of $ \Lambda^{2}T\left(T^{*\left(-1\right)}Z\right) $) in
the last three factors vanish. Indeed,
before the anti-symmetrization
in the formula
$$
\left[\eta,\widetilde\omega\right]=\operatornamewithlimits{Alt}_{i j m}\eta^{il} \widetilde\omega_{,l}^{jm}
$$
only members with $ j $, $ m $ in the
direction of fibers (i.e., $ j,m\geq k+1 $) and $ l $ in the direction of the
base (i.e., $ l\leq k $) remain, hence
only members with $ i $, $ j $, $ m $ in the direction of the fibers
remain. The
anti-symmetrization evidently preserves this property.
To bring the prove to an end we can note that the resulting formula
for $ \left[\eta,\widetilde\omega\right] $
$$
\left[\eta,\widetilde\omega\right] = \operatornamewithlimits{Alt}\Sb i jm \\ i,j,m\geq k+1\endSb\sum_{l\leq k}\left(\delta^{i+k,l}-\delta^{i-k,l}\right)\widetilde\omega_{,l}^{jm}=-\operatornamewithlimits{Alt}\Sb i jm \\ i,j,m\geq k+1\endSb\widetilde\omega_{,i-k}^{jm}
$$
coincides with the formula for exterior differentiation up to a
sign
and that the generalization to the case of a cotangent bundle to a
foliation doesn't meet any obstacles.\qed\enddemo
\head Double cohomology of a double complex \endhead
As we have already seen, the classification problem for
bihamiltonian manifolds with a given reduction (which is a
Veroneze web) is reduced to a linear problem: find all the solutions
of
$$
d_{\lambda}d_{\mu}t=0,\qquad t\in\Gamma\left(T^{*\left(-1\right)}X\right),
$$
modulo
$$
t= d_{\lambda}\varphi_{1}+d_{\mu}\varphi_{2},\qquad \varphi_{1},\varphi_{2}\in\Gamma\left(\Lambda^{0}T^{*\left(-1\right)}X\right)\simeq\Gamma\left({\Cal O}\right).
$$
\proclaim{ Hypothesis } For a point $ x $ on the Veroneze web $ X $ there exists
a neighborhood $ U $ such that
$$
{\Bbb H}^{i}\left(U\right)=0\quad \text{for }i\geq1.
$$
Here $ {\Bbb H}^{i} $ denotes the double cohomology of the double complex of
sub-differential forms on $ X $.
\endproclaim
In the case of analytical manifolds category we know a proof of
this hypothesis. However, in the case of $ C^{\infty} $-category the
situation concerning many cohomological invariants is, as we know,
quite different comparing with analytical case. So in the former
case it is better to call this hypothesis a question.
Anyway, in the case of analytical manifolds this hypothesis
(the proof of which we will write elsewhere) allows as to prove
the following
\proclaim{ Theorem } An analytical bihamiltonian manifold of odd dimension in
general position is defined locally by its bihamiltonian reduction
(which is a Veroneze web) up to a local diffeomorphism. \endproclaim
\head Cited literature \endhead
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structures related to them. Func. Anal. Appl., 13, p.~248--262 (1979).
4. Gelfand~I.~M., Zakharevich~I.~S. Spectral theory for a pair of
skew-symmetrical operators on $ S^{1} $. Func. Anal. Appl., 23, p.~85--93
(1989).
5. Gelfand~I.~M., Dorfman~I. Hamiltonian operators and the
classical Yang---Baxter equation. Func. Anal. Appl., Vol. 16 (1982),
pp.~241--248.
6. Arnold~V.~I. The mathematical methods of classical mechanics.
Springer-Verlag, New-York, 1978.
7. Kroneker~L. Algebraische Reduction der Schaaren bilinearen
Formen. Sitzungber. Acad. Berlin, 1890, pp.~763--776. Mathematische
Werke, B.~3(2), p.~139--155; Chelsea, New-York, 1968.
\enddocument