# Difference between revisions of "Algebraic Geometry Seminar Fall 2011"

The seminar meets on Fridays at 2:25 pm in Van Vleck B215.

The schedule for the previous semester is here.

## Fall 2011

date speaker title host(s)
Sep. 23 Yifeng Liu (Columbia) Enhanced Grothendieck's operations and base change theorem for

sheaves on Artin stacks

Tonghai Yang
Sep. 30 Matthew Ballard (UW-Madison) You got your Hodge Conjecture in my matrix factorizations (local)
Oct. 7 Zhiwei Yun (MIT) Cohomology of Hilbert schemes of singular curves Shamgar Gurevich
Oct. 14 Javier Fernández de Bobadilla (Instituto de Ciencias Matematicas, Madrid) Nash problem for surfaces L. Maxim
Oct. 21 Andrei Caldararu (UW-Madison) The Hodge theorem as a derived self-intersection (local)
Nov. 11 John Francis (Northwestern) Integral transforms and Drinfeld centers in derived algebraic geometry Andrei Caldararu
Nov. 18 Sukhendu Mehrotra (Madison) Moduli spaces of sheaves on K3 surfaces and generalized deformations (local)
Dec. 2 Shamgar Gurevich (Madison) Canonical Hilbert Space: Why? How? and its Categorification
Dec. 9 Sean Paul (Madison) Semistable pairs and quasi-closed orbits

## Spring 2012

date speaker title host(s)
March 16 Weizhe Zheng (Columbia) TBD Tonghai Yang
March 23 Ryan Grady (Notre Dame) Twisted differential operators as observables in QFT. Caldararu
May 4 Mark Andrea de Cataldo (Stony Brook) TBA Maxim

## Abstracts

TBA

### Matthew Ballard

You got your Hodge Conjecture in my matrix factorizations

Abstract: I will describe how to prove some new cases of Hodge conjecture using the following tools: categories of graded matrix factorizations, the homotopy category of dg-categories, Orlov's Calabi-Yau/Landau-Ginzburg correspondence, Kuznetsov's relationship between the derived categories of a certain K3 surface and the Fermat cubic fourfold, and Hochschild homology. This is joint work with David Favero (Wien) and Ludmil Katzarkov (Miami/Wien).

### Zhiwei Yun

Cohomology of Hilbert schemes of singular curves

Abstract: For a smooth curve, the Hilbert schemes are just symmetric powers of the curve, and their cohomology is easily computed by the H^1 of the curve. This is known as Macdonald's formula. In joint work with Davesh Maulik, we generalize this formula to curves with planar singularities (which was conjectured by L.Migliorini). In the singular case, the compactified Jacobian will play an important role in the formula, and we make use of Ngo's technique in his celebrated proof of the fundamental lemma.

Nash problem for surfaces

The space of arcs through the singular set of an algebraic variety has a infinite dimensional scheme structure. In the late sixties Nash proved that it has finitely many irreducible components. He defined a natural mapping from this set of irreducible components to the set of essential divisors of a resolution of singularities. Roughly speaking the set of essential divisors is the set of irreducible components of the exceptional divisor of a resolution whose birational transform is an irreducible component of the exceptional divisor of any other resolution.

Nash proved that this mapping is injective and proposed to study its bijectivity. In 2003 S. Ishii and J. Kollar gave a counterexample to the surjectivity in dimension at least 4. Recently, in a joint work with M. Pe Pereira, the speaker has settled the bijectivity for surfaces. In this talk I will explain the proof.

### Andrei Caldararu

The Hodge theorem as a derived self-intersection

The Hodge theorem is one of the most important results in complex geometry. It asserts that for a complex projective variety X the topological invariants H^*(X, C) can be refined to new ones that reflect the complex structure. The traditional statement and proof of the Hodge theorem are analytic. Given the multiple applications of the Hodge theorem in algebraic geometry, for many years it has been a major challenge to eliminate this analytic aspect and to obtain a purely algebraic proof of the Hodge theorem. An algebraic formulation of the Hodge theorem has been known since Grothendieck's work in the early 1970's. However, the first purely algebraic (and very surprising) proof was obtained only in 1991 by Deligne and Illusie, using methods involving reduction to characteristic p. In my talk I shall try to explain their ideas, and how recent developments in the field of derived algebraic geometry make their proof more geometric.

### John Francis

Integral transforms and Drinfeld centers in derived algebraic geometry

For a finite group G, conjugation invariant vector bundles on G have a universal property with respect to Rep(G): they form its Drinfeld center. Joint work with David Ben-Zvi and David Nadler generalizes this result, extending work of Hinich, in the setting of derived algebraic geometry. We describe a generalization of the Drinfeld center for a monoidal stable infinity category as a Hochschild cohomology category. For quasi-coherent sheaves on a perfect stack X, we prove that its center is equivalent to sheaves on the derived loop space LX. The structure of this category of sheaves defines an extended 2-dimensional topological quantum field theory.

### Sukhendu Mehrotra

Moduli spaces of sheaves on K3 surfaces and generalized deformations

It is a result of Mukai that connected components of the moduli space of stable sheaves on a K3 surface X are holomorphic symplectic varities. As any such component Y deforms in a 21 dimensional family, while the moduli space of K3 surfaces is 20 dimensional, the general deformation Y' of Y will not be be a moduli space of sheaves on a K3. This talk presents an attempt to associate to such a Y' a "non-commutative K3 surface" $\displaystyle{ \hat{X} }$ for which the modular description carries over. This joint work with Eyal Markman.