Seminari / Porta e Mazzon
SEMINARIO DI GEOMETRIA ALGEBRICA
Nell'ambito del seminario congiunto di geometria algebrica organizzato dai Dipartimenti di Matematica dell'Università e del Politecnico di Milano,
lunedì 4 giugno alle ore 14:30 e 15:30
presso l'Aula C del Dipartimento di Matematica dell'Università di Milano si terranno i seguenti seminari:
Mauro Porta (ore 14:30)
(Université de Strasbourg)
HKR theorem in analytic geometry
Abstract: In this talk I will survey recent advances in derived geometry that allow to obtain a complex and rigid analytic version of the Hochschild-Kostant-Rosenberg theorem.
Classically, this theorem states that Hochschild homology (the algebraic counterpart of the self-intersection of the diagonal) is isomorphic to the de Rham algebra.
Using techniques from derived geometry, Toën and Vezzosi lifted this isomorphism to an equivalence at the chain level, and showed it to be multiplicative and to be compatible with the natural circle action on the Hochschild side and the de Rham differential on the differential side.
In this talk I will explain the motivations that lead us to seek for an analytic HKR theorem. After, I'll give a new simplified proof of the HKR theorem in the algebraic setting, and I'll explain why this new proof can be generalized to the complex and rigid analytic setting.
This is joint work with J. Antonio and F. Petit.
Enrica Mazzon (ore 15:30)
Berkovich approach to degenerations of hyper-Kähler varieties
Abstract: To a degeneration of varieties, we can associate the dual intersection complex, a topological space that encodes the combinatoric of the central fiber and reflects the geometry of the generic fiber. In this talk I will show how the techniques of Berkovich geometry give an insight in the study of the dual complexes. In this way, we are able to determine the homeomorphism type of the dual complex of some degenerations of hyper-Kähler varieties. The results are in accordance with the predictions of mirror symmetry, and the recent work about the rational homology of dual complexes of degenerations of hyper-Kähler varieties, due to Kollár, Laza, Saccà and Voisin. This is joint work with Morgan Brown.