Algebraic Analysis and Geometry Monday, 4 July 2011 - Padova - Italy

Schedule

Abstracts

Ettore Aldrovandi, Mapping spaces of group-like stacks

Stacks with a group law (gr-stacks, for short) arise, for example, as automorphisms of algebraic or topological stacks. Morphisms between them are essentially derived mapping spaces between two length-one complexes of non necessarily abelian groups. I will show how they can be computed by certain diagrams called butterflies. In the abelian case this reproduces a previous result by Deligne characterizing the derived category in terms of morphisms of Picard stacks. The main applications are to the long exact sequence and to the change of coefficient map in nonabelian cohomology. This is based on joint work with Behrang Noohi.

François Charles, The standard conjectures for some holomorphic symplectic varieties

Holomorphic symplectic varieties are smooth simply connected projective varieties which carry a unique everywhere nondegenerate two-form. Using techniques from hyperkähler geometry, we prove Grothendieck's standard conjectures in the theory of algebraic cycles for those varieties which are deformations of Hilbert schemes of points on K3 surfaces. This is joint work with Eyal Markman.

Ajay Ramadoss, A variant of the Mukai pairing via deformation quantization

We use an algebraic index theorem of P. Bressler, R. Nest and B. Tsygan to give a relatively short computation of a certain pairing on the Hochschild homology of a smooth projective variety. This pairing is closely related to the Mukai pairing constructed by A. Caldararu.

Antonio Rapagnetta, Moduli spaces of sheaves on K3 or abelian surfaces

Irreducible symplectic holomorphic manifolds play a central role in the classification of compact kaehler varieties with trivial canonical bundle. Up to deformation equivalence very few examples are known. About 25 years ago Mukai suggested to construct examples starting from moduli spaces of sheaves on projective surfaces with trivial canonical bundle. We will focus on the geometry of the irreducible symplectic holomorphic manifolds that can be obtained by desingularizing singular moduli spaces of sheaves on those surfaces.

Carlo Rossi, The UEA (Universal Enveloping Algebra) of an $L_\infty$-algebra via deformation quantization

The UEA of an $L_\infty$-algebra has been constructed by Baranovsky using methods of homological perturbation theory. We propose here a different insight using deformation quantization: the (graded) Formality theorem of Kontsevich and the Formality Theorem in presence of two branes permit to prove that indeed deformation quantization of an $L_\infty$ algebra is quasi-isomorphic to the construction of Baranovsky, but admits a more explicit construction, permits to prove a PBW Theorem and to construct an explicit quasi-isomorphism between both constructions, reminiscent of the Duflo element in the theory of finite-dimensional Lie algebras.

(Last update: Tuesday May 3, 2022)

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