Lie Theory Workshop, U. C. Berkeley, February 25-26, 2017

Transformation Groups in Complex Geometry

Conference in Honor of Alan Huckleberry

Titles and Abstracts

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Daniel Barlet Meromorphic quotients for some holomorphic G-actions

Abstract: Using mainly tools from references [B.13] and [B.15] we try to make a first step toward a "Transcendental Geometric Invariant Theory", that is to say to study conditions for the existence of "meromorphic quotients" for a holomorphic actions of a complex Lie group G on a reduced complex space X. In this article we give necessary and sufficient conditions [H.1], [H.2] and [H.3] on the G-orbits' configuration in X, in order that a holomorphic action of a connected complex Lie group G on a reduced complex space X admits a "strongly quasi-proper meromorphic quotient". Under these conditions a canonical (minimal) such quotient exists and it factorizes in a canonical way any G-invariant meromorphic map defined on X. In order to show how these conditions can be used, we apply this characterization to obtain that, when G = K.B with B a closed complex subgroup of G and K a real compact subgroup of G, the existence of a strongly quasi-proper meromorphic quotient for the B-action implies the existence of a strongly quasi-proper meromorphic quotient for the G-action on X, assuming moreover that the action of B on X satisfies the condition [H.1str] (a strong version of [H.1]) on a G-invariant dense subset, we prove also that this last condition is automatically satisfied for G when K normalizes B and when [H.1str], [H.2] and [H.3] are satisfied for B. We also give a similar result when the connected complex Lie group has the form G = K.A.K where A is a closed connected complex subgroup and K is a compact (real) subgroup, assuming that the A-action satisfies the hypothesis [H.1str] on a G-invariant open set U_1, the hypothesis [H.2] on a G-invariant open set U_0 in U_1, and [H.3]. We prove the existence of a natural holomorphic map between the two meromorphic quotients of X for the actions of B and G (resp. of A and G) when they exist, and we discuss the properness of this map.

The references are:

[B.13] Barlet, D. Quasi-proper meromorphic equivalence relations, Math. Z. (2013), vol. 273, p. 461-484.

[B.15] Barlet, D. Strongly quasi-proper maps and f-flattning theorem, math.arXiv.1504.01579v1 [math. CV].

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Ana-Maria Brecan Cycle transversal Mumford-Tate domain Abstract: Flag domains are open orbits of real semisimple Lie groups in flag manifolds of their complexifications. A special class of flag domains constitute the classifying spaces for variations of Hodge structure, namely period domains or more generally Mumford-Tate domains. In this talk I will consider the problem of classifying all equivariant embeddings of an arbitrary flag domain in a period domain satisfying a certain transversality condition. Satake studied this problem in the weight 1 case in connection to the study of (algebraic) families of abelian varieties where the transversality condition is trivial. In this talk I will describe certain combinatorial structures at the Lie algebra level, called Hodge triples, which are generalisation of sl(2)-triples and show how this structures provide a solution to the classification problem.
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Peter Heinzner Convex hulls of momentum map images We will describe the faces of the convex hull of a restricted momentum map defined on a compact Kaehler manifold $X$. Restricted momentum maps control the the action of a given real form $G_0$ on $X$. We show that the faces of the covex hull of its image are exposed and define in a natural way parabolic subgroups of $G_0$. We will discuss the geometry of their action in some detail.
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Jaehyun Hong Smooth horospherical varieties of Picard number one Abstract: A homogeneous space G/H is said to be horospherical if G is reductive and H contains the unipotent radical of a Borel subgroup of G. In this case the normalizer P of H in G is parabolic and the morphism from G/H to G/P is a torus bundle over a flag variety. A horospherical variety is a normal G-variety having an open dense G-orbit which is horospherical. For example, flag varieties and toric varieties are horospherical varieties. In this talk we investigate complex geometry of smooth horospherical varieties of Picard number one: classifications, automorphism groups, geometric structures, characterizations by varieties of minimal rational tangents.
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Alan Huckleberry On complex geometric curvature of flag domains

Abstract: Flag domains are complex manifolds which arise as open orbits of real Lie groups in flag manifolds of their complexifications. They play an important role in various contexts, including in the theory of parameter spaces of complex structures on compact couplex manifolds (period domains, Hodge theory, Mumford-Tate domains) and in the representation theory of semisimple Lie groups. The lecture will focus on flag domains from the complex analytic viewpoint. Our main goal will be to sketch two different types of results (joint with T. Hayama and Q. Latif and with J. Hong and A. Seo) which contribute to understanding the pseudoconcavity of a flag domain $D$. If $D$ possesses non-constant holomorphic functions, then it is a classical fact that it is the product of a compact flag manifold and Hermtian symmetric space of non-compact type. In particular we show that otherwise, i.e., if $D$ possesses only the constant holomorphic functions, then it is pseudoconcave. We also give lower bounds on the degree of of the pseudoconcavity which are explicitly computed in many cases in terms of root theoretic invariants. It is expected that this will lead to finiteness theorems of cohomology in low degree by adapting methods of Andreotti and Grauert to our setting.

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Colleen Robles Characterization of Gross's Calabi-Yau variations of Hodge structure type by characteristic forms

Abstract: Gross showed that to every Hermitian symmetric tube domain we may associate a canonical variation of Hodge structure (VHS) of Calabi-Yau type. The construction is representation theoretic, not geometric, in nature, and it is an open question to realize this abstract VHS as the variation induced by a family of polarized, algebraic Calabi-Yau manifolds.

In order for a geometric VHS to realize Gross's VHS it is necessary that the invariants associated to the two VHS coincide. For example, the Hodge numbers must agree. The later are discrete/integer invariants. Characteristic forms are differential-geometric invariants associated to VHS (introduced by Sheng and Zuo).

Remarkably, agreement of the characteristic forms is both necessary and sufficient for a geometric VHS to realize one of Gross's VHS. That is, the characteristic forms characterize Gross's Calabi-Yau VHS. I will explain this result, and discuss how characteristic forms have been used to study candidate geometric realizations of Gross's VHS.

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Domingo Toledo Period Domains and Fundamental Groups of Varieties Abstract: The purpose of this talk is to bring attention to the following problem. Let G be a simple, non-compact, linear Lie group that has a compact Cartan subgroup but whose symmetric space is not Hermitian symmetric. For example, G = SO(2p,q) for p > 1, q > 2. Let \Gamma be a (torsion-free, co-compact) lattice in G. Can \Gamma be the fundamental group of a smooth projective variety? If D = G/V is a period domain associated to G, and X is the quotient of D by \Gamma, then \Gamma is the fundamental group of the non-algebraic manifold X. It is known that the topology of X is far from that of an algebraic manifold. But it is not known if there could be a smooth projective variety with the same fundamental group as X. We will discuss this problem and related problems, and explain how they are related to the geometry of period domains.
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