Geometry and Analysis Seminar

A number of moduli problems are, via Hodge theory, closely related to 
ball quotients. In this situation there is often a choice of possible 
compactifications such as the GIT compactification´and its Kirwan 
blow-up or the Baily-Borel compactification and the toroidal 
compactificatikon. The relationship between these compactifications is 
subtle and often geometrically interesting. In this talk I will discuss 
several cases, including cubic surfaces and threefolds and 
Deligne-Mostow varieties. This discussion links several areas such as 
birational geometry, moduli spaces of pointed curves, modular forms and 
derived geometry. This talk is based on joint work with S. 
Casalaina-Martin, S. Grushevsky, S. Kondo, R. Laza and Y. Maeda.

A well-known problem in algebraic geometry is to construct smooth projective Calabi--Yau varieties $Y$. In the smoothing approach, we construct first a degenerate (reducible) Calabi--Yau scheme $V$ by gluing pieces. Then we aim to find a family $f\colon X \to C$ with special fiber $X_0 = f^{-1}(0) \cong V$ and smooth general fiber $X_t = f^{-1}(t)$. In this talk, we see how infinitesimal logarithmic deformation theory solves the second step of this approach: the construction of a family out of a degenerate fiber $V$. This is achieved via the logarithmic Bogomolov--Tian--Todorov theorem as well as its variant for pairs of a log Calabi--Yau space $f_0\colon X_0 \to S_0$ and a line bundle $\mathcal{L}_0$ on $X_0$.
 

I will present a notion of spin structure on a perfect complex in characteristic zero, generalizing the classical notion for an (algebraic) vector bundle. For a complex $E$ on $X$ with an oriented quadratic structure one obtains an associated ${\mathbb Z}/2{\mathbb Z}$-gerbe over X which obstructs the existence of a spin structure on $E$. This situation arises naturally on moduli spaces of coherent sheaves on Calabi-Yau fourfolds. Using spin structures as orientation data, we construct a categorical refinement of a K-theory class constructed by Oh-Thomas on such moduli spaces.

Bryant’s Laplacian flow is an analogue of Ricci flow that seeks to flow an arbitrary initial closed $G_2$-structure on a 7-manifold toward a torsion-free one, to obtain a Ricci-flat metric with holonomy $G_2$. This talk will give an overview of joint work with Mark Haskins and Rowan Juneman about complete self-similar solutions on the anti-self-dual bundles of ${\mathbb CP}^2$ and $S^4$, with cohomogeneity one actions by SU(3) and Sp(2) respectively. We exhibit examples of all three classes of soliton (steady, expander and shrinker) that are asymptotically conical. In the steady case these form a 1-parameter family, with a complete soliton with exponential volume growth at the boundary of the family. All complete Sp(2)-invariant expanders are asymptotically conical, but in the SU(3)-invariant case there appears to be a boundary of complete expanders with doubly exponential volume growth.

This talk reports on two projects. The first work (in progress), joint  with Amanda Hirschi, constructs (genus 0) open Gromov-Witten invariants for any Lagrangian submanifold using a global Kuranishi chart construction. As an application we show open Gromov-Witten invariants are invariant under Lagrangian cobordisms. I will then describe how open Gromov-Witten invariants fit into mirror symmetry, which brings me to the second project: obtaining open Gromov-Witten invariants from the Fukaya category.

I will discuss recent and ongoing work with Davesh Maulik that explains how Gromov-Witten invariants behave under simple normal crossings degenerations. The main outcome of the study is that if a projective manifold $X$ undergoes a simple normal crossings degeneration, the Gromov-Witten theory of $X$ is determined, via universal formulas, by the Gromov-Witten theory of the strata of the degeneration. Although the proof proceeds via logarithmic geometry, the statement involves only traditional Gromov-Witten cycles. Indeed, one consequence is a folklore conjecture of Abramovich-Wise, that logarithmic Gromov-Witten theory “does not contain new invariants”. I will also discuss applications of this to a conjecture of Levine and Pandharipande, concerning the relationship between Gromov-Witten theory and the cohomology of the moduli space of curves.

It is a long-standing open problem to generalize sheaf-counting invariants of complex projective three-folds to symplectic manifolds of real dimension six. One approach to this problem involves counting  J-holomorphic curves  C, for a generic almost complex structure J, with weights depending on J. Various existing symplectic invariants (Gromov-Witten, Gopakumar-Vafa, Bai-Swaminathan) can be expressed as such weighted counts. In this talk, based on joint work with Thomas Walpuski, I will discuss a new construction of weights associated with curves and a closely related problem about the structure of the space of Cauchy-Riemann operators on  C.

I will discuss how some ideas from Geometric Langlands can be used to obtain new results in birational geometry and on the topology of algebraic varieties.