Volume 10 (2010), Number 1. Abstracts

D. Arinkin and R. Bezrukavnikov. Perverse Coherent Sheaves [PDF]

This note introduces an analogue of perverse t-structure on the derived category of coherent sheaves on an algebraic stack (subject to some mild technical conditions). Under additional assumptions construction of coherent “intersection cohomology” sheaves is given. Those latter assumptions are rather restrictive but hold in some examples of interest in representation theory.

Similar results were obtained by Deligne (unpublished), Gabber and Kashiwara.

Keywords. Coherent perverse sheaves, coherent IC sheaves.

2000 Mathematics Subject Classification. Primary: 18F20; Secondary: 14A20, 14F05.

D. Calaque and M. Van den Bergh. Global Formality at the G-Level [PDF]

In this paper we prove that the sheaf of L-polydifferential operators for a locally free Lie algebroid L is formal when viewed as a sheaf of G-algebras via Tamarkin's morphism of DG-operads GB.

In an appendix we prove a strengthening of Halbout's globalization result for Tamarkin's local quasi-isomorphism.

Keywords. Deformation quantization.

2000 Mathematics Subject Classification. Primary: 14F99, 14D99.

D. Gaitsgory and D. Nadler. Spherical Varieties and Langlands Duality [PDF]

Let G be a connected reductive complex algebraic group. This paper is devoted to the space Z of meromorphic quasimaps from a curve into an affine spherical G-variety X. The space Z may be thought of as a finite-dimensional algebraic model for the loop space of X. The theory we develop associates to X a connected reductive complex algebraic subgroup \check H of the dual group \check G. The construction of \check H is via Tannakian formalism: we identify a certain tensor category Q(Z) of perverse sheaves on Z with the category of finite-dimensional representations of \check H. The group \check H encodes many aspects of the geometry of X.

Keywords. Loop spaces, Langlands duality, quasimaps.

2000 Mathematics Subject Classification. Primary: 22E67; Secondary: 14H60, 55P35.

A. Goncharov. Hodge correlators II [PDF]

We define Hodge correlators for a compact Kähler manifold X. They are complex numbers which can be obtained by perturbative series expansion of a certain Feynman integral which we assign to X. We show that they define a functorial real mixed Hodge structure on the rational homotopy type of X.

The Hodge correlators provide a canonical linear map from the cyclic homomogy of the cohomology algebra of X to the complex numbers.

If X is a regular projective algebraic variety over a field k, we define, assuming the motivic formalism, motivic correlators of X. Given an embedding of k into complex numbers, their periods are the Hodge correlators of the obtained complex manifold.

Motivic correlators lie in the motivic coalgebra of the field k. They come togerther with an explicit formula for their coproduct in the motivic Lie coalgebra.

Keywords. Mixed Hodge structure, cyclic homology, Feynman integral.

2000 Mathematics Subject Classification. 14.

U. Jannsen and M. Rovinsky. Smooth Representations and Sheaves [PDF]

The paper is concerned with a ‘geometrization’ of smooth representations of the automorphism group of universal domains, and with the properties of ‘geometric’ representations of such groups. As an application, we calculate the cohomology groups of several classes of smooth representations of the automorphism group of an algebraically closed extension of infinite transcendence degree of an algebraically closed field.

Keywords. Automorphism groups of fields, smooth representations, Grothendieck topologies.

2000 Mathematics Subject Classification. 12F20, 12G99, 14E99, 18E15, 18F10, 18F20, 43A65.

S. Loktev. Weight Multiplicity Polynomials of Multi-Variable Weyl Modules [PDF]

This paper is based on the observation that dimension of weight spaces of multi-variable Weyl modules depends polynomially on the highest weight. We support this conjecture by various explicit answers for up to three variable cases and discuss the underlying combinatorics.

Keywords. Current algebra, Weyl module.

2000 Mathematics Subject Classification. 17B65, 17B10.

T. Terasoma. DG-categories and Simplicial Bar Complexes [PDF]

We prove that the DG category KCA of DG complexes in CA assocaited to a DGA A, is homotopy equivalent to that of comodules over the bar complex of A. We introduce simplicial bar complexes to give the homotopy equivalence. As an application, we show that the category of comodules over the 0-th cohomology of the bar complex of the Deligne algebra is equivalent to that of variations of mixed Tate Hodge structures on an algebraic variety.

Keywords. Bar complex, DG-category, Deligne cohomology.

2000 Mathematics Subject Classification. 14F43, 14F45.

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