Conference speakers and abstracts

• Nick Addington (University of Oregon)

Hermitian-Einstein connections on universal bundles

For a smooth, compact moduli space of stable vector bundles on a K3 surface, we construct a Hermitian-Einstein connection on the universal bundle, which restricts to a H-E connection on its "wrong-way" slices. Thus not only does the moduli space parametrize stable bundles on the K3 surface, but the K3 surface parametrizes stable bundles on the moduli space. Ingredients include the Bismut-Freed-Quillen connection on a determinant line bundle, and a careful analysis of the curvature of a certain principal bundle coming from gauge theory. This is joint work with Andrew Wray.

• Ben Bakker (University of Illinois at Chicago)

Period integrals of algebraic varieties

Period integrals on complex algebraic varieties are the integrals of algebraic differential forms along topological cycles. They are at the heart of Hodge theory. In this talk I will survey some recent results on the behavior of the functions obtained by taking period integrals in algebraic families, including an Ax--Schanuel type theorem on their transcendence and the relationship to the Grothendieck period conjecture. I will also include some geometric applications, including at least one to hyperkahler manifolds. This is joint work with J. Tsimerman.

• Arend Bayer (University of Edinburgh)

Kuznetsov categories of Fano threefolds

I will give a survey of recent results on Kuznetsov categories of Fano threefolds of Picard rank one. These results give additional structure to their classification, and their moduli spaces. The techniques involved include moduli spaces of Bridgeland-stable objects, Brill-Noether statements, and equivariant categories (spiced with a pinch of derived algebraic geometry).

• Izzet Coskun (University of Illinois at Chicago)

The cohomology of a general tensor product of stable bundles on the plane

In this talk, I will explain how to compute the cohomology of the tensor product of two general stable bundles on the plane. Let V and W be two stable bundles general in their moduli. Assume that the numerical invariants of W are sufficiently divisible. Using recent advances in the Minimal Model Program for moduli spaces, we completely compute the cohomology of the tensor product of V and W. In particular, when W is exceptional, we show that the tensor product of V and W has at most one nonzero cohomology group, generalizing foundational results of Drézet, Göttsche and Hirschowitz. We also characterize when the tensor product of V and W is globally generated. This is joint work with Jack Huizenga and John Kopper.

• Laure Flapan (Michigan State University)

Kodaira dimension of some moduli spaces of hyperkähler manifolds

We study the geometry of some moduli spaces of polarized hyperkähler manifolds. We use techniques of Gritsenko-Hulek-Sankaran involving the Borcherds modular form to determine a bound on the degree of the polarization after which these moduli spaces are always of general type. This is joint work with I. Barros, P. Beri, and E. Brakkee.

• Daniel Halpern-Leistner (Cornell University)

The noncommutative minimal model program

There are many situations in which the derived category of coherent sheaves on a smooth projective variety admits a semiorthogonal decomposition that reflects something interesting about its geometry. I will present a new unifying framework for studying these semiorthogonal decompositions using Bridgeland stability conditions. Then, I will formulate some conjectures about canonical flows on the space of stability conditions that imply several important conjectures on the structure of derived categories, such as the D-equivalence conjecture and Dubrovin's conjecture.

• Max Lieblich (University of Washington)

Moduli spaces of cohomology classes

I will discuss various ways in which cohomology groups can act as moduli spaces and ways in which moduli stacks can carry universal cohomology classes. Some of what I discuss is joint work with Daniel Bragg and some is joint work with Sheela Devadas.

• Yuchen Liu (Northwestern University)

Moduli of K3 surfaces and K-stability

Classically, there are two approaches to compactify moduli spaces of K3 surfaces: GIT (geometric) and Baily-Borel compactification (Hodge theory). In this talk, I will explain how K-stability can be used to construct new moduli spaces for certain K3 surfaces, especially in degree 2 and degree 4, where these K-moduli spaces and their wall crossings naturally interpolate between the GIT and Baily-Borel. As an application, we verify a conjecture of Laza and O’Grady. Based on joint works with Kenneth Ascher and Kristin DeVleming.

• Eyal Markman (University of Massachusetts Amherst)

Rational Hodge isometries of hyper-Kahler varieties of K3[n]-type are algebraic

Let X and Y be compact hyper-Kahler manifolds deformation equivalence to the Hilbert scheme of length n subschemes of a K3 surface. A cohomology class in their product XxY is an analytic correspondence, if it belongs to the subalgebra generated by Chern classes of coherent analytic sheaves. Let f be a Hodge isometry of the second rational cohomologies of X and Y with respect to the Beauville-Bogomolov-Fujiki pairings. We prove that f is induced by an analytic correspondence. We furthermore lift f to an analytic correspondence F between their total rational cohomologies, which is a Hodge isometry with respect to the Mukai pairings, and which preserves the gradings up to sign. When X and Y are projective the correspondences f and F are algebraic.

• Tony Pantev (University of Pennsylvania)

Homotopy complex structures, doubling, and shifted symplectic geometry

I will describe a symplectic approach to generalized complex geometry which by design produces a natural category of gc branes (at the large volume limit) and allows us to define and work with gc branes with substrates which are not submanifolds. The approach utilizes shifted symplectic geometry and a new theory of homotopy complex structures on derived stacks. I will discuss briefly this theory and will explain its implications for gc geometry. This is a report of a joint work in progress with P. Safronov and B. Pym and on recent work of Y. Qin.

• Laura Pertusi (University of Milan)

Uniqueness of dg enhancements for Kuznetsov components

The study of the bounded derived category of coherent sheaves on a smooth projective variety is a central topic in algebraic geometry. Almost all the functors arising from geometric constructions are of Fourier-Mukai type, namely they can be described by an object in the derived category of a product. In this setting, Orlov proved in 1996 that every exact fully faithful functor with adjoints is of FM type. Since then, this result has been further generalized and a useful tool is to enhance the triangulated structure on the derived category to a dg category.

In this talk we consider certain admissible subcategories of the bounded derived category of cubic fourfolds, Gushel-Mukai varieties and quartic double solids, known as Kuznetsov components, and we show the strongly uniqueness of their dg enhancement making use of stability conditions with special properties. As application, we show that equivalences among the above mentioned Kuznetsov components are of FM type. This is the content of a joint work with Chunyi Li and Xiaolei Zhao.

• Aaron Pixton (University of Michigan)

Double-double ramification cycles

The double ramification (DR) cycle is a codimension g cycle on the moduli space of stable genus g curves C with marked points p_i, corresponding to a single divisorial condition O_C(a_1*p_1 + ... + a_n*p_n) ~ O_C. The double-double ramification cycle (DDR) is a codimension 2g cycle corresponding to taking two such divisorial conditions. I'll first explain why the DDR cycle is not just the product of two DR cycles. I'll then describe how to define and (in principle) compute DDR cycles by working on certain iterated blowups of the moduli space of stable curves. This is joint work with D. Holmes, S. Molcho, R. Pandharipande, and J. Schmitt.

• Junliang Shen (Yale University)

Fourier-Mukai transforms and the decomposition theorem for integrable systems

For an integrable system associated with a holomorphic symplectic variety (i.e. a Lagrangian fibration), we consider two natural geometric structures attached to it: (1) the duality of the derived category of coherent sheaves induced by the Fourier-Mukai transform, and (2) the decomposition theorem of Beilinson-Bernstein-Deligne. We present a proposal that relates these two structures using Saito’s theory of Hodge modules, and discuss connections to the “perverse = Hodge” symmetry, Matsushita’s higher direct image theorem, and homological mirror symmetry for hyper-holomorphic sheaves. Based on joint work in progress with Davesh Maulik and Qizheng Yin.

• Evgeny Shinder (University of Sheffield)

Derived elliptic structures on K3 surfaces

We define a derived elliptic structure on a K3 surface X as an elliptic fibration on a Fourier-Mukai partner of X. I explain how to classify derived elliptic structures in terms of the Brauer group and the discriminant lattice of X, and show that in general derived elliptic structures do not arise from relative Jacobians of elliptic structures on X itself. This gives a negative answer to a 2014 question by Hassett and Tschinkel. This is joint work in progress with Reinder Meinsma.

• Yukinobu Toda (Kavli IPMU)

Categorical and K-theoretic Donaldson-Thomas theory of C^3

The generating series of Donaldson-Thomas invariants associated with Hilbert schemes of points on the three-dimensional affine space is known to form a MacMahon function, whose coefficients are numbers of plane partitions. In this talk, I will give a categorical and K-theoretic analogue of the above formula. I will consider the triangulated categories of matrix factorizations of super-potentials whose critical loci are Hilbert schemes of points, and show the existence of their semi-orthogonal decompositions which are regarded as categorification of the above MacMahon formula. In fact there exist explicitly constructed objects in each semi-orthogonal summand, whose cardinality is the number of plane partitions, which generate the torus localized K-group of the above category of matrix factorizations. This is a joint work with Tudor Padurariu.