Abstracts

Amin Gholampour

University of Maryland

August 6 — 2:50 pm

Stable Sheaves and Pairs on Calabi-Yau 4-folds

In the first part of the talk that is a joint work in progress with Yunfeng Jiang and Jason Lo, I will discuss a 2-dimensional stable pair theory for 4-folds parallel to Pandharipande-Thomas 1-dimensional stable pair theory of 3-folds. In Calabi-Yau case, we apply Oh-Thomas theory to define invariants counting these stable pairs under some restrains. I will talk about some examples and applications including local surfaces and threefolds. If time allows in the second part of the talk, I will discuss applying Oh-Thomas theory to obtain a virtual cycle for the (fixed locus of) the moduli spaces of sheaves on some class of threefolds with a torus action.  

Ravindra Girivaru

University of Missouri, St. Louis

August 5 — 1:00 pm

Some Lefschetz Theorems for Vector Bundles.

The theme of this talk is the following: Given a smooth, projective variety Y and a smooth hyperplane section X in Y, we explore when a vector bundle on X extends to Y. In the case when Y is the projective space, additional hypotheses on the vector bundle and the hyperplane section yields splitting theorems for the vector bundles being studied.

Martijn Kool

Utrecht University

August 5 — 9:00 am

Virtual Invariants of Projective Surfaces

Moduli spaces of stable sheaves on general type surfaces are typically singular. However, they carry a virtual class which can be used to define intersection numbers such as virtual Euler and Segre numbers. The former are part of SU(r) Vafa-Witten invariants. I give an overview of these invariants highlighting recent developments, such as a mathematical definition of SU(r)/Z_r Vafa-Witten invariants in terms of twisted sheaves, a new formula for SU(5) Vafa-Witten invariants in terms of the Rogers-Ramanujan continued fraction, and a virtual Segre-Verlinde correspondence. Joint works with Goettsche, Goettsche-Laarakker, and Jiang.

Justin Lacini

University of Kansas

August 7 — 9:00 am

Logarithmic Bounds on Fujita's Conjecture

A longstanding conjecture of T. Fujita asserts that if X is a smooth complex projective variety of dimension n and if L is an ample line bundle, then K_X+mL is basepoint free for m>=n+1. The conjecture is known up to dimension five by work of Reider, Ein, Lazarsfeld, Kawamata, Ye and Zhu. In higher dimensions, breakthrough work of Angehrn, Siu, Helmke and others showed that the conjecture holds if m is larger than a quadratic function in n. We show that for n>=2 the conjecture holds for m larger than n(loglog(n)+3). This is joint work with L. Ghidelli.

Yuchen Liu

Princeton University

August 5 — 10:50 am

Finite Generation and K-stability

K-stability was first introduced by Tian to characterize the solution of the Kähler-Einstein problem on Fano varieties. In the last decade, a purely algebro-geometric study of K-stability has prospered. An important outcome of this theory is the construction of a well-behaved moduli space of Fano varieties. In this talk, we will introduce the higher rank finite generation (HRFG) conjecture, and explain its connection to several foundational questions in K-stability. Then we will discuss the solution of the higher rank finite generation conjecture, based on recent joint work with Chenyang Xu and Ziquan Zhuang.

Jason Lo

California State University, Northridge

August 7 — 10:50 am

Autoequivalences, Stability, and Elliptic Surfaces

The study of autoequivalences of a triangulated category has applications in mirror symmetry, categorical dynamical systems, moduli spaces and their counting invariants. In this talk, I will describe a connection between two results - one in representation theory and one in algebraic geometry - where stability is preserved under an autoequivalence of a triangulated category. I will then explain how this connection leads to new understanding of the stability manifold of an elliptic surface.

Jayan Mukherjee

University of Kansas

August 5 — 2:50 pm

Deformations of Quadruple Canonical Covers  and the Moduli of Surfaces of General Type

In this article we study the  deformations of the canonical morphism $\varphi:X\to \mathbb{P}^N$ of irregular surfaces $X$ of general type with at worst canonical singularities, when $\varphi$ is a finite Galois morphism of degree $4$ onto a smooth variety of minimal degree $Y$ inside $\mathbb{P}^N$. These surfaces satisfy $K_X^2 = 4p_g(X)-8$, with $p_g$ an even integer, $p_g \geq 4$. They are classified in \cite{GP} into four distinct families (three, if $p_g=4$). We show that, when $X$ is general in its family, any deformation of $\varphi$ has degree greater than or equal to $2$ onto its image. More interestingly, we prove that, with two exceptions, a general deformation of $\varphi$ is two--to--one onto its image, which is a surface whose normalization is a ruled surface of appropriate genus. We also show that with the exception of one family, the deformations of a general surface $X$ are unobstructed, and consequently, $X$ belongs to a unique irreducible component of the Gieseker moduli space, which we prove is uniruled (the fourth one being product of curves is well-studied). As a consequence, we show the existence of infinitely many moduli spaces with uniruled components corresponding to each even $p_g\geq 4$. Among other things, our results are relevant because they exhibit moduli components such that the degree of the canonical morphism jumps up at proper locally closed subloci. This contrasts with the moduli of surfaces with $K_X^2 = 2p_g - 4$ (which are double covers of surfaces of minimal degree),studied by Horikawa but is pleasingly similar to the moduli of curves of genus $g\geq 3$.

Ikshu Neithalath

University of Southern Denmark

August 6 — 1:00 pm

Sheaf-theoretic SL(2,C) Floer Homology for Knots and their Surgeries

SL(2,C) Floer homology is an invariant of 3-manifolds (resp. knots) defined by Abouzaid-Manolescu (resp. Cote-Manolescu) that applies objects from algebraic geometry, such as character varieties and perverse sheaves of vanishing cycles, to build an invariant in low-dimensional topology. We will discuss joint work with Cote on the properties of the knot invariant as well as independent work computing the 3-manifold invariant for surgeries on some knots.

Song Sun

University of California, Berkeley

August 7 — 1:00 pm

Reflexive Sheaves, Hermitian-Yang-Mills Connections, and Tangent Cones

The Donaldson-Uhlenbeck-Yau theorem relates the existence of Hermitian-Yang-Mills connections over a compact Kahler manifold with algebraic stability of a holomorphic vector bundle. This has been extended by Bando-Siu in 1994 to a class of singular Hermitian-Yang-Mills connections on reflexive sheaves. We study tangent cones of these singular connections in the geometric analytic sense, and show that they can be characterized in terms of new algebro-geometric invariants of reflexive sheaves. Based on joint work with Xuemiao Chen (University of Maryland). 

Thomas Walpuski

Humboldt-Universität zu Berlin

August 6 — 9:00 am

The Gopakumar-Vafa Finiteness Conjecture

In 1998, using arguments from M–theory, Gopakumar and Vafa argued that there are integer BPS invariants of symplectic Calabi–Yau 3–folds. Unfortunately, they did not give a direct mathematical definition of their BPS invariants, but they predicted that they are related to the Gromov–Witten invariants by a transformation of the generating series. The Gopakumar–Vafa conjecture asserts that if one defines the BPS invariants indirectly through this procedure, then they satisfy an integrality and a (genus) finiteness condition.

The integrality conjecture has been resolved by Ionel and Parker. A key innovation of their proof is the introduction of the cluster formalism: an ingenious device to side-step questions regarding multiple covers and super-rigidity. Their argument could not resolve the finiteness conjecture, however. The reason for this is that it relies on Gromov’s compactness theorem for pseudo-holomorphic maps which requires an a priori genus bound. It turns out, however, that rather powerful tools from geometric measure theory imply a compactness theorem for pseudo-holomorphic cycles. This can be used to upgrade Ionel and Parker’s cluster formalism and prove both the integrality and finiteness conjecture.

This talk is based on joint work with Eleny Ionel and Aleksander Doan.

Yuanqi Wang

University of Kansas

August 6 — 10:50 am

The Spectrum of a Dirac Operator Associated to Singular G_2-instantons 

We establish a relation between the spectrum of a Dirac operator on S5 and certain sheaf cohomologies on the complex projective plane. This operator comes from the deformation of G_2-instantons with 1-dim singularities.