Algebraic Geometry Seminar

Department of Mathematics
Johns Hopkins University

Fall 2016

Regular meeting time: Tuesdays 4:30-5:30 (Tea served at 4:00)
Place: Shaffer 202

Date Speaker Title
September 20 Steven Zucker
A tale of two Hodge structures.
The so-called Zucker conjecture from 1980 was proved by others in two very different ways in 1987. It asserts that the L^2 cohomology of a locally symmetric variety (with respect to its natural metric) is isomorphic to the middle perversity intersection cohomology of its Baily-Borel compactification. This provides two ways to provide a Hodge structure on the latter space: (1) via L^2 harmonic differential forms, (2) via Mo. Saito's mixed Hodge modules, equivalently the more geometric construction of de Cataldo and Migliorini. There is no reason, a priori, why these should coincide, or even have the same Hodge numbers. Nonetheless, it is expected that they do. I will present my approach to proving that. It would please the representation theorists to know that their L^2 automorphic forms are filtered the right way!
October 4 Yunqing Tang
Institute for Advanced Study
Cycles in the de Rham cohomology of abelian varieties over number fields
In his 1982 paper, Ogus defined a class of cycles in the de Rham cohomology of smooth projective varieties over number fields. In the case of abelian varieties, this class includes all the Hodge cycles by the work of Deligne, Ogus and Blasius. Ogus predicted that all such cycles are Hodge. This conjecture is a crystalline analogue of the Mumford-Tate conjecture. In this talk, I will discuss the proof of Ogus' conjecture for some families of abelian varieties under the assumption that the cycles lie in the Betti cohomology with real coefficients. The proof is based on known cases of the Mumford-Tate conjecture and a theorem of Bost on algebraic foliation.
October 18 Lei Wu
Northwestern University
Multi-indexed Deligne extensions and a Kawamata-Viehweg type vanishing for Log- VHS
Log D-modules are natural generalization of D-modules for a pair (X, D) consisting of a complex manifold X and a normal crossing divisor D. I will construct a special kind of log D-modules, called multi-indexed Deligne extensions. The construction uses the idea of Bernstein-Sato polynomials. As an application, I will prove a Kawamata-Viehweg type vanishing for Log-variations of Hodge structures.
November 1 Heather Lee
Institute for Advanced Study
Homological mirror symmetry for open Riemann surfaces from pair-of-pants decompositions
We will demonstrate one direction of HMS for punctured Riemann surfaces -- the wrapped Fukaya category of a punctured Riemann surface is equivalent to the matrix factorization category MF(X,W) of the toric Landau-Ginzburg mirror (X, W).
The category MF(X,W) can be constructed from a Cech cover of (X,W) by local affine pieces that are mirrors of pairs of pants. We supply a suitable model for the wrapped Fukaya category for a punctured Rimemann surface so that it can also be explicitly computed in a sheaf-theoretic way, from the wrapped Fukaya categories of various pairs of pants in a decomposition. The pieces are glued together in the sense that the restrictions of the wrapped Floer complexes from two adjacent pairs of pants to their adjoining cylindrical piece agree.
November 15, 2-3 pm Krieger 413 Yu-jong Tzeng
University of Minnesota
New relations on Caporaso-Harris invariants and more
Caporaso-Harris invariants count the number of nodal curves on the projective plane satisfying given tangency conditions with a line. Those invariants are first proved to satisfy a recursive formula in 1996 and then showed to be polynomials in the degree (when the degree is large enough) using tropical geometry in 2009. On the other hand, the number of nodal curves are already known to be universal polynomials of the Chern numbers for any surfaces and line bundles. In this talk we will discuss a new type of relations for Caporaso-Harris invariants which unify these features and in fact also hold on all smooth varieties.
November 15, 4:30-5:30 pm Lawrence Ein
University of Illinois at Chicago
Measures of irrationality
We 'll discuss joint work with Bastianelli, De Poi, Lazarsfeld and Ullery. We study various measure of irrationality.
In particular, we study the case, when our variety X is a general hypersurface of degree d in P^{n+1} and deg X > 2n.
December 6 Alex Perry
Columbia University
Categorical joins
Homological projective duality is a powerful theory developed by Kuznetsov for studying the derived categories of varieties. It can be thought of as a categorification of classical projective duality. I will describe a categorification of the classical join of two projective varieties, its relation to homological projective duality, and some applications. This is joint work with Alexander Kuznetsov.