Algebraic Analysis and Geometry Monday, 21 June 2010 - Padova - Italy

Schedule

Abstracts

Julien Grivaux, Hochschild classes for coherent analytic sheaves and the Grothendieck-Riemann-Roch theorem

The Grothendieck-Riemann-Roch theorem is a fundamental result in algebraic geometry; it describes how the Chern character for coherent sheaves behaves under proper direct images. On abstract complex manifolds, a theory of characteristic classes for coherent analytic sheaves can still be developed in any reasonable cohomology theory, but the GRR theorem is only known in Dolbeault and De Rham cohomology. In this talk, I will explain a new proof of the GRR theorem in Dolbeault cohomology, based on an approach through Hochschild homology due to Kashiwara.

Ernesto Mistretta, Bigness of vector bundles, Chern classes and semistability

(joint work, in progress, with Simone Diverio)
Starting from some hyperbolicity questions, we give a differential criterion sufficient to provide bigness for a hermitian holomorphic vector bundle on a complex manifold. We give then a (still conjectural) version of such criterion in term of semistability and some inequalities on Chern classes, together with some evidences. As a byproduct we show how these techniques lead to another proof and a geometrical interpretation of the classical Kobayashi-Lübke inequality

Etienne Mann, Mirror symmetry for weigthed projective spaces via D-modules

We first describe a canonical mirror partner (B-model) of the small quantum orbifold cohomology of weighted projective spaces (A-model) in the framework of differential equations: we attach to the A-model (resp. B-model) a D-module on the torus and we show that these two D-modules are isomorphic.

François Petit, A Riemann-Roch Theorem for dg Algebras

Let A be a proper, homologically smooth, differential graded algebra. Given a perfect A-module M and an endomorphism f of M, one defines the Hochschild class of a pair (M,f) with values in the Hochschild homology of A. We will show that this class commutes with "convolution" and explains the link with the Riemann-Roch theorem for differential graded algebras.

Stéphane Guillermou, Quantization of Hamiltonian isotopies and non displaceability

(joint work with Masaki Kashiwara and Pierre Schapira)
We consider a homogeneous symplectic transformation Φ of the cotangent bundle of some variety M with the zero-section removed. If Φ is in the connected component of the identity map and satisfies some properness hypothesis we show that there exists a sheaf on M×M whose microsupport is the graph of Φ (up to an antipodal transformation). We deduce some already known versions of Arnold's conjectures about the "non-displaceability" of some subsets of the cotangent bundle.

(Last update: Tuesday June 1, 2010)

---