Algebraic geometry seminar

from Tuesday
January 01, 2019 to Friday
May 31, 2019
Show events for:
Instructions for subscribing to Stony Brook Math Department Calendars

Wednesday
January 30, 2019

4:00 PM - 5:30 PM
Math Tower P-131
Samuel Grushevsky, Stony Brook University
Cohomology of compactifications of moduli of cubic threefolds

The moduli space of complex cubic threefolds admits various compactifications, by viewing it as a GIT quotient, a ball quotient, or via the intermediate Jacobians. We compute and compare the (intersection) cohomology of various compactifications. Based on joint work with S. Casalaina-Martin, K. Hulek, and R. Laza


Wednesday
February 06, 2019

4:00 PM - 5:30 PM
Math Tower P-131
Misha Verbitsky, IMPA/ HSE
Automorphisms of algebraically hyperbolic manifolds

A projective manifold M is called "algebraically hyperbolic" if there exists a positive constant A such that the degree of any curve of genus g on M is bounded from above by A(g−1).

It is not hard to see that Kobayashi hyperbolicity implies algebraic hyperbolicity. It is known that Kobayashi hyperbolic manifolds have finite automorphism groups. We have shown that, more generally, algebraically hyperbolic projective manifolds have finite automorphism groups. This is a joint work with Fedor Bogomolov and Ljudmila Kamenova.


Wednesday
February 13, 2019

4:00 PM - 5:30 PM
Math Tower P-131
Dawei Chen, Boston College / IAS
Volumes and intersection theory on moduli spaces of abelian differentials

Computing volumes of moduli spaces has significance in many fields. For instance, the celebrated Witten’s conjecture regarding intersection numbers on the Deligne-Mumford moduli space of stable curves has a fascinating connection to the Weil-Petersson volume, which motivated Mirzakhani to give a proof via Teichmueller theory, hyperbolic geometry, and symplectic geometry. The initial two other proofs of Witten’s conjecture by Kontsevich and by Okounkov-Pandharipande also used various ideas in ribbon graphs, Gromov-Witten theory, and Hurwitz theory. In this talk I will introduce an analogue of Witten’s intersection numbers, defined on the Bainbridge-Chen-Gendron-Grushevsky-Moeller compactification of moduli spaces of Abelian differentials, that can be used to compute the Masur-Veech volumes. This is joint work with Moeller, Sauvaget, and Zagier (arXiv:1901.01785).


Wednesday
February 20, 2019

4:00 PM - 5:30 PM
Math Tower P-131
Junliang Shen, MIT
Perverse filtrations, Gopakumar-Vafa invariants, and hyper-kähler geometry

For a hyper-kähler variety equipped with a Lagrangian fibration, an increasing filtration is defined on its rational cohomology using the perverse t-structure. We will discuss the role played by this filtration in the study of the topology and geometry of hyper-kähler varieties, as well as the connection to curve counting invariants of Calabi-Yau 3-folds. In particular, we will discuss some recent progress on the P=W conjecture for Hitchin systems, and its compact analog for Lagrangian fibrations. Based on joint work with Qizheng Yin and Zili Zhang.


Wednesday
February 27, 2019

4:00 PM - 5:30 PM
Math Tower P-131
Benjamin Bakker, University of Georgia
Hodge theory and o-minimality

The cohomology groups of complex algebraic varieties come equipped with a powerful invariant called a Hodge structure. Going back to foundational work of Griffiths, Hodge theory has found many important applications to algebraic and arithmetic geometry, but its intrinsically analytic nature often leads to complications. Recent joint work with Y. Brunebarbe, B. Klingler, and J. Tsimerman has shown that in fact many Hodge-theoretic constructions can be carried out in an intermediate geometric category, and o-minimality provides the crucial tameness hypothesis to make this precise. In this talk I will describe how this perspective can be used to easily recover an important theorem of Cattani--Deligne--Kaplan on the algebraicity of Hodge loci and to prove a conjecture of Griffiths on the quasiprojectivity of the images of period maps.


Wednesday
March 06, 2019

4:00 PM - 5:30 PM
Math Tower P-131
Sophie Morel, Princeton
Relative Nori motives (joint with F. Ivorra)

The classical category of Nori motives is the "smallest abelian category" through which the relative Betti cohomology functors (X,Y,n)->H^n(X,Y) factor; here X is a variety over C, Y is a closed subvariety and n is an integer. If S is a base variety over C, or more generally over any field of characteristic 0, there are two ways to extend this approach to get a category of Nori motives over S : the first, due to Arapura, uses ordinary sheaf cohomology, and the second, due to Ivorra, uses perverse sheaf cohomology. Although the second approach seems more complicated, it turns to be more amenable to the construction of motivic analogues of the sheaf operators (such as direct image by a morphism of varieties); the reason is that many of these operations are known to be exact from the perverse point of view, and that Beilinson has given recipes to get the other operations from the exact ones. We also are able to get a theory of weights without using Saito's mixed Hodge modules. As an application, we recover results of the type "the cohomology of objects of geometric origin has a mixed Hodge structures that is compatible with maps of geometric origin" that were originally obtained by Zucker. M. Saito, de Cataldo and de Cataldo-Migliorini.


Wednesday
March 13, 2019

4:00 PM - 5:30 PM
Math Tower P-131
Dan Abramovich, Brown University
Resolving singularities in families

Semistable reduction is often the first step in constructing compactified moduli spaces, and can be used to discover their properties. I will describe work-in-progress with Michael Temkin and Jaroslaw Wlodarczyk in which we prove functorial semistable reduction for families of varieties in characteristic 0, refining work with Karu from 2000. Techniques developed for moduli spaces enter in unexpected ways.


Monday
March 25, 2019

4:00 PM - 5:00 PM
Math Tower P-131
Olivier Martin, University of Chicago
Measures of irrationality for very general abelian varieties

In recent years renewed attention has been brought to measures of irrationality for projective varieties. While vector bundle methods have been leveraged by Bastianelli, de Poi, Ein, Lazarsfeld, and Ullery to study the degree of irrationality and covering gonality of high degree hypersurfaces, Voisin has used rational equivalence of zero-cycles to show that the covering gonality of a very general abelian variety of dimension g goes to infinity with g. I will sketch how one can generalize Voisin's method in order to prove the following conjecture: A very general abelian variety of dimension at least 2k-1 has covering gonality greater than k. If time permits, I will explain how this method can be used to obtain new lower bounds on the degree of uni-irrationality of abelian varieties.


Wednesday
April 03, 2019

2:30 PM - 3:30 PM
Math P-131
Ignacio Barros, Northeastern University
Two moduli spaces of Calabi-Yau type.

(SPECIAL TIME, BUT BACK IN THE USUAL PLACE)

For fixed genus $g$, the transition of $\mathcal{M}_{g,n}$ from uniruled to general type as $n$ increases is rather sudden, making the cases of intermediate Kodaira dimension very rare. In genus $4≤ g ≤ 11$ there are just a few cases for which we still don't know the Kodaira dimension of $\mathcal{M}_{g,n}$, some of them with very rich geometries. I will report on work-in-progress with Scott Mullane, where we study some of this cases hoping to find more instances of intermediate Kodaira dimension.


Wednesday
April 24, 2019

4:00 PM - 5:30 PM
Math Tower P-131
Valery Alexeev, University of Georgia
Degenerations of K3 surfaces and 24 points on the sphere

I will discuss Kulikov and stable degenerations of K3 surfaces and describe explicit, geometric compactifications of their moduli spaces in several interesting cases. Based on joint work with Philip Engel and Alan Thompson.


Wednesday
May 01, 2019

4:00 PM - 5:30 PM
Math Tower P-131
Andrew Obus, Baruch College CUNY
Fun with Mac Lane valuations

Mac Lane's technique of "inductive valuations" is over 80 years old, but has only recently been used to attack problems about arithmetic surfaces. We will give an explicit, hands-on introduction to the theory, requiring little background beyond the definition of a valuation on a ring. We will then outline how this theory is helpful for resolving "weak wild" quotient singularities of arithmetic surfaces, as well as for proving conductor-discriminant inequalities for higher genus curves. The first project is joint with Stefan Wewers, and the second is joint work with Padmavathi Srinivasan.


Monday
May 06, 2019

4:00 PM - 5:00 PM
Math Tower P-131
Geoff Smith, Harvard University
TBA


Monday
May 06, 2019

4:00 PM - 5:00 PM
Math Tower P-131
Geoff Smith, Harvard
Low degree points on curves

Faltings' theorem asserts that a smooth curve C of genus at least 2 over a number field K has only finitely many rational points. Given this fact, one can consider the union of C(e) of all the sets C(L), where L is any extension of K of degree at most some fixed integer e, and ask whether C(e) is finite. It is easy to see that if C(e) is finite, than e is less than the gonality of C over K. This provides an “expected" answer to the question. In this talk I will show that the question of finiteness of C(e) has this expected answer for smooth curves cut out by a sufficiently ample class on a surface of irregularity 0. To do so, I will use a result of Faltings to convert the question into one about the behavior of effective line bundles on C over an algebraic closure of K, then address that question with tools from complex algebraic geometry. This work was joint with Isabel Vogt.


Wednesday
May 08, 2019

4:00 PM - 5:30 PM
Math Tower P-131
Laura Schaposnik, UIC and SCGP
Higgs bundles, branes, and applications

Higgs bundles are pairs of holomorphic vector bundles and holomorphic 1-forms taking values in the endomorphisms of the bundle. Their moduli spaces carry a natural Hyperkahler structure, through which one can study Lagrangian subspaces (A-branes) or holomorphic subspaces (B-branes). Notably, these A and B-branes have gained significant attention in string theory. After introducing Higgs bundles and the associated Hitchin fibration, we shall look at natural constructions of families of different types of branes during the first part of the talk. The later part of the talk will be dedicated to relating these spaces to the study of 3-manifolds, surface group representations and mirror symmetry.


Show events for:
Instructions for subscribing to Stony Brook Math Department Calendars