Titles & Abstracts

Title: Algebraic groups as automorphism groups of projective varieties 

Abstract: It is known that every connected real Lie group can be realized as the full automorphism group of some Stein, complex hyperbolic complex manifold. The talk will discuss the analogous question of realizing a connected algebraic group as the full automorphism group of some projective algebraic variety. 

Title: Geometry of Hermitian symmetric spaces under the action of a maximal unipotent group 

Abstract: Given a complex manifold M with a Lie group G action by holomorphic transformations, it is of interest to understand associated invariant objects like the invariant Stein subdomains and the invariant plurisubharmonic functions. A classical example of this framework is given by tube domains in complex Euclidean space, where M=C^n and G=R^n acts by translations.

In this talk, we present a generalization of the results about tube domains in the setting of a Hermitian symmetric space of the non-compact type G/K under the action of a maximal unipotent subgroup N of G.  

This is work in collaboration with Andrea Iannuzzi.

Title: Remembrances of Dima

Abstract: In October of 1985 we began almost 40 years of interaction. The personal and mathematical can not be separated.

Title: From the crown domain to invariant Hyperkähler complexifications of bounded homogeneous domains.  

Abstract: In 1990 Dmitri N. Akhiezer and Simon G. Gindikin detected a particular Kähler complexification D of an arbitrary Riemannian symmetric space G/K which turned out to be relevant in different contexts, e.g. complex and Kähler geometry, harmonic analysis, representation theory. Firstly, we would like to recall some related results arising from a question posed by Dmitri Akhiezer in 2009. Time permitting, we will also indicate how the special features of D in the case of G/K Hermitian symmetric may lead, via an explicit Lie theoretic construction, to a novel invariant Hyperkähler complexification of an arbitrary homogeneous bounded domain in C^n (work in progress).  

Title: Factorization of holomorphic matrices

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Title: Momentum maps and the Kähler property for base spaces of reductive principal bundles

Abstract: An arbitrary holomorphic action of a complex reductive group G on a Kähler manifold can be rather pathological. The situation is much better, however, if there is a momentum map for the action of a maximal compact subgroup of G. In this talk I will report on a recent joint result with Daniel Greb where we considered the case that G acts freely and properly with compact quotient and looked for conditions under which the quotient manifold is again Kähler. 

Title: Orbits and invariants related to coisotropy representations 

Abstract: Let G/H be a quasi-affine homogeneous space, G being semisimple. I am going to discuss old and new results related to the coisotropy representation of H. In particular, I'll present some complements to results of my joint paper with Dmitri Akhiezer  "Multiplicities in the branching rules and the complexity of homogeneous spaces".  

Title: Graded covering of a supermanifold 

Abstract: In geometry there is a well-known notion of a covering space. A classical example is the following universal covering: p : R → S^1, p(t)=exp(it). Another example of this notion is a flat covering or torsion-free covering in the theory of modules over a ring. All these coverings satisfy some common universal properties. In the paper "Super Atiyah classes and obstructions to splitting of supermoduli space'', Donagi and Witten suggested a construction of a first obstruction class for splitting a supermanifold. It appeared that an infinite prolongation of the Donagi-Witten construction satisfies universal properties of a covering. In other words this is a covering of a supermanifold in the category of graded manifolds. Our talk is devoted to the notion of a covering in the super Lie theory and supergeometry. 

Title: Coisotropic actions and complete integrability 

Abstract: A coadjoint orbit Qx of an algebraic group Q is a symplectic variety. These orbits provide a natural setting for integrable systems. Following the usual terminology, we say that an algebra A of regular functions on Qx is complete, if the  Lie—Poisson  bracket vanishes on it, i.e., {A,A}=0, and the transcendence degree of A is equal to dim(Qx)/2. We will explain, why the Gelfand—Tsetlin subalgebras are complete on all adjoint orbits. 

The talk is based on a joint work with D.I. Panyushev.

Title: Gromov’s ellipticity: old and new

Abstract: The classical principle of Oka-Grauert asserts that the classification in continuous category of vector bundles and, more generally, fiber bundles with homogeneous fibers and linear algebraic structure groups over Stein complex spaces, resp. classification of their sections up to homotopy, coincide with similar classifications in analytic category. This principle was significantly generalized by M. Gromov and then further extended by F. Forstnerič, F. Lárusson and others. Moreover, Gromov suggested an extension of this principle to algebraic category. 

We will stay with Gromov’s notion of ellipticity in the algebraic setting and survey some old and very recent results in the subject.