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.
|Sep. 23||Yifeng Liu (Columbia)||Enhanced Grothendieck's operations and base change theorem for
sheaves on Artin stacks
|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|
|Dec. 2||Shamgar Gurevich (Madison)||Canonical Hilbert Space: Why? How? and its Categorification|
|Dec. 9||Sean Paul (Madison)||Hyperdiscriminants and Semistable pairs||(local)|
|March 2||Marti Lahoz (Bonn)||TBD||Mehrotra|
|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|
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).
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.
Javier Fernández de Bobadilla
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.
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.
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.
Canonical Hilbert space: Why? How? and its Categorification
There is an idea in the mathematical physics community that quantization should be a functorial procedure. Our motivation in this talk is to show an example of such a procedure, answering a question of Kazhdan. I will describe a natural construction of an explicit quantization functor from the category SYMP of finite-dimensional symplectic vector spaces over a finite field to the category VECT of finite-dimensional complex vector spaces. In particular, for a fixed symplectic vector space V we obtain a canonical Hilbert space H(V), acted upon by the symplectic group Sp(V). This is called the canonical model of the Weil representation.
The main idea in the construction of our functor is to overcome the traditional choice of a Lagrangian that appears in the classical constructions in the field of geometric quantization. For doing this we will explain the Grothendieck geometrization procedure, replacing sets by algebraic varieties, and function theoretic constructions, by sheaf theoretic analogues. In particular, I will explain the use of "Perverse Extension" to improve on the standard constructions that appear in the literature.
Time permits, I will explain the categorification, or sign problem, which appears naturally in our setting. This categorification problem was formulated by Bernstein and Deligne, and was solved recently with the help of Ofer Gabber (IHES). I will speak on it in the future.
Joint work with Ronny Hadani (Austin).
I will assume only knowledge of very elementary representation theory.
Sean Timothy Paul
Hyperdiscriminants and Semistable Pairs
One of the main problems in complex geometry is to detect the existence of "canonical" Kahler metrics in a given Kahler class on a compact complex (Kahler) manifold. In particular one seeks necessary and sufficient conditions for the existence of a Kahler Einstein metric on a Fano manifold. In this case the presence of positive curvature makes this problem extremely difficult and has led to a striking series of conjectures--the "standard conjectures"-- which relate the existence of these special metrics (which are solutions to the complex Monge-Ampere equation, a fully non-linear elliptic p.d.e . ) to the algebraic geometry of the pluri-anticanonical images of the manifold. Yau speculated that the relevant algebraic geometry would be related (somehow) to Mumford's Geometric Invariant Theory. Eventually it was conjectured that K-energy bounds along Bergman potentials could be deduced from an appropriate notion of "semi-stability". Recently this conjecture has been completely justified by the speaker, building upon work of Gang Tian and Gelfand-Kapranov-Zelevinsky and Weyman-Zelevinsky. It is the aim of this talk to outline progress on the standard conjectures and to discuss the entire theory in the context of complex algebraic groups and dominance of rational representations of such groups.