All lectures in HG G43 [HWZ]
10:45-12:00 Shen I
10:45-12:00 Shen II
10:45-12:00 Shen III
Titles and Abstracts
Title: Multiple covers and point counts for K3 surfaces
Abstract: The Gromov-Witten theory of K3 surfaces in arbitrary curve classes is conjecturally governed by quasi-modular forms and should satisfy holomorphic anomaly equations. The primitive case is well understood whereas the higher divisibility case is significantly more complicated. We explain the multiple cover conjecture and discuss partial results in divisibility 2. The counts of genus g curves passing through g generic points will serve as a leading example. The talk is based on joint work in progress with Younghan Bae.
Title: The curious hard Lefschetz property for character varieties
Abstract: I will talk about a way to decompose the character variety of a Riemann surface of arbitrary rank with prescribed semisimple generic local monodromies into cells where each cell looks like a product of an affine space and a symplectic torus. This can be thought of as abelianization. As an application, we deduce the curious hard Lefschetz property conjectured by Hausel, Letellier and Rodriguez-Villegas, which claims that the operator of cup product with the class of the holomorphic symplectic form is an isomorphism between complementary degrees of the associated graded with respect to the weight filtration on the cohomology.
Title: Motivic decompositions of the Hilbert scheme of points of a K3 surface
Abstract: I will discuss joint work with Andrei Negut and Qizheng Yin in which we construct an explicit, multiplicative Chow-Kunneth decomposition of any Hilbert scheme of points of a K3 surface. This is parallel to the case of abelian varieties where such a decomposition is induced by the Fourier-Mukai transform. Our approach relies on a lift of the Looijenga-Lunts-Verbitsky Lie algebra to Chow.
Title: Gushel-Mukai varieties and stability conditions
Abstract: generic Gushel-Mukai variety X is a quadric section of a linear
section of the Grassmannian Gr(2,5). Kuznetsov and Perry proved that the bounded derived category of X has a semiorthogonal decomposition with exceptional objects and a non-trivial subcategory Ku(X), known as the Kuznetsov component. In this talk we will discuss the construction of stability conditions on Ku(X) and, consequently, on the bounded derived category of X. As applications, for X of even-dimension, we will construct locally complete families of hyperkaehler manifolds from moduli spaces of stable objects in Ku(X) and we will determine when X
has a homological associated K3 surface. This is a joint work with Alex Perry and Xiaolei Zhao.
Title: Base loci of big and nef line bundles on irreducible symplectic varieties
Abstract: In the first part of this talk, I give a complete description of the divisorial part of the base locus of big and nef line bundles on irreducible symplectic varieties (under certain conditions). This is a generalization of well-known results of Mayer and Saint-Donat for K3 surfaces. In the second part, I will present what is currently known on the non-divisorial part, including the results of an ongoing cooperation with Daniele Agostini.
Title: Almost three definitions of double ramification cycles and why they are equivalent
Abstract: Given a smooth projective curve C and distinct points p1,...,pn in C,
one can study the condition that there exists a meromorphic k-differential form on C with zeros and poles at the points p1, ..., pn of given orders. The double ramification cycles are cohomology classes on the moduli space of stable curves (C,p1,...,pn) associated to extending this condition to all stable curves. Different approaches have been proposed for defining these cycles. I will introduce some of them and discuss how work in progress with Bae, Holmes, Pandharipande and Schwarz finishes the proof that these give the same cohomology classes.
Title: Hitchin systems, compact hyper-Kaehler geometry, and the P=W conjecture
Abstract: The topology of Hitchin systems play a central role in geometry, mathematical physics, and representation theory.
In 2010 , de Cataldo, Hausel, and Migliorini predict a surprising connection between the topology of Hitchin systems and the Hodge theory of character varieties, which is now referred to as the P=W conjecture.
In the three lectures, we will discuss structures of the cohomology of the moduli of Higgs bundles, recent progress on the P=W conjecture, and compact analogs of the P=W conjecture. We will present a proof of the genus 2 case for arbitrary rank, and a proof of the P=W for the even tautological ring for arbitrary genus. Our strategy is to use hidden symmetries in the compact hyper-Kaehler geometry and a degeneration method connecting the compact geometry and Hitchin systems. If time permits, we will also discuss connections to curve counting invariants and some open questions.
Title: Hodge structures of hyperkahler manifolds
Abstract: Cohomology of a compact hyperkähler manifold carries a natural Lie algebra action introduced by Verbitsky and Looijenga-Lunts. In my talk, I will discuss some applications of this fact. First, one can use it to recover the Hodge structures on higher degree cohomology groups from the Hodge structure in degree two. Second, it controls the monodromy action on the cohomology in smooth families of hyperkähler manifolds. The latter observation can be used to study the behaviour of Hodge structures under degeneration.
Title: A compact analogue of P=W
Abstract: I will present the proof of a compact version of P=W, which relates the topology of holomorphic Lagrangian fibrations to the Hodge theory of degenerations of compact hyper-KÃ¤hler manifolds. Hopefully this reveals certain hyper-KÃ¤hler nature of the P=W phenomenon. I will also discuss possible ways to categorify the statement. Joint work with Andrew Harder, Zhiyuan Li, and Junliang Shen.