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||Andrei Caldararu (UW-Madison)||The Hodge theorem as a derived self-intersection||(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|
|Nov. 25||Shamgar Gurevich (Madison)||Canonical Hilbert Space: Why? How? and its Categorification
|May 4||Mark Andrea de Cataldo (Stony Brook)||TBA||Maxim|
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.
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.