## Moduli Spaces in Algebraic Geometry

##### Session Organizers

Alexander Polishchuk, University of Oregon

### 8 December, 2022

##### Abstract

Harder-Narasimhan (HN) theory gives a structure theorem for principal G bundles on a smooth projective curve. A bundle is either semistable, or it admits a canonical filtration whose associated graded bundle is semistable in a graded sense. After reviewing recent advances in extending HN theory to arbitrary algebraic stacks, I will discuss work in progress with Andres Fernandez Herrero to apply this general machinery to the stack of “gauged” maps from a curve C to a G-scheme X, where G is a reductive group and X is projective over an affine scheme. Our main application is to use HN theory for gauged maps to compute generating functions for K-theoretic enumerative invariants known as gauged Gromov-Witten invariants. This problem is interesting more broadly because it can be formulated as an example of an infinite dimensional analog of the usual set up of geometric invariant theory, which has applications to other moduli problems.

##### Abstract

I this talk, I will give semiorthogonal decompositions of derived categories of several classical moduli spaces, e.g. symmetric products of curves, Brill-Noether loci, (relative) Quot schemes, Hilbert schemes of points. In particular, they contain a proof of Jiang’s conjecture for semiorthogonal decompositions of Quot schemes of locally free quotients. They are by-products of my research on categorifications of wall-crossing in Donaldson-Thomas theory, and the proofs involve techniques of derived algebraic geometry, categorical Hall algebras, matrix factorizations and Koszul duality.

##### Abstract

We study non-commutative projective varieties in the sense of Artin-Zhang, which are given by non-commutative homogeneous coordinate rings, which are finite over their centre. We construct moduli spaces of stable modules for these, and construct a symmetric obstruction theory in the CY3-case. This gives deformation invariants of Donaldson-Thomas type. The simplest example is the Fermat quintic in quantum projective space, where the coordinates commute up to carefully chosen 5th roots of unity. We explore the moduli theory of finite length modules, which mixes features of the Hilbert scheme of commutative 3-folds, and the representation theory of quivers with potential. This is joint work with Yu-Hsiang Liu, with contributions by Atsushi Kanazawa.

##### Abstract

I will report on joint work with Daniel Bragg. There are many ways to express the universality properties of moduli spaces. For example, Vakil established years ago that any singularity type defined over the integers appears in natural moduli spaces, proving what he called “Murphy’s Law”. We have discovered another, similar, phenomenon: many natural moduli spaces contain all finite gerbes. In particular, any finite gerbe over any field appears as the residual gerbe of some point of the stack of curves.

### 9 December, 2022

##### Abstract

The moduli stack of complexes of vector bundles over a Gorenstein Calabi-Yau curve, considered up to chain isomorphisms, admits a 0-shifted Poisson structure in the sense of Calaque-Pantev-Toen-Vaquie-Vezzosi. We will give several classical examples of Poisson varieties appearing in representation theory and integrable system, that are naturally Poisson substacks of the above mentioned stack. Using derive algebraic geometry, we are able to prove some new results on these Poisson varieties. This is a joint work with Alexander Polishchuk.

##### Abstract

The construction of stability conditions on the bounded derived category of coherent sheaves on smooth projective varieties is notoriously a difficult problem. In this talk, I will review some results and techniques related to this problem. I will specifically concentrate on the case of Hilbert schemes of points on K3 surfaces and on generic abelian varieties of any dimension. This is joint work in progress with C. Li, E. Macrì and P. Stellari.

##### Abstract

We consider tautological bundles and their exterior and symmetric powers over the Quot scheme of zero dimensional quotients over the projective line. We prove several results regarding the vanishing of their higher cohomology, and we describe the spaces of global sections via tautological constructions. This is based on joint work with Alina Marian, Shubham Sinha and Steven Sam.

##### Abstract

I will discuss joint work with Kenny Ascher, Dori Bejleri, Harold Blum, Giovanni Inchiostro, Yuchen Liu, and Xiaowei Wang on construction of moduli stacks and moduli spaces of log Calabi Yau pairs that can be realized as slc log Fano pairs with complements. Unlike moduli of canonically polarized varieties (respectively, Fano varieties) in which the moduli stack of KSB stable (respectively, K semistable) objects is bounded for fixed volume, dimension, the objects here form unbounded families. Despite this unbounded behavior, in the case of plane curve pairs (P2, C), we construct a projective good moduli space parameterizing S-equivalence classes of these slc Fanos with complements.

##### Abstract

Grothendieck’s Existence Theorem asserts that a coherent sheaf on a scheme proper over a complete local noetherian ring is the same as a compatible system of coherent sheaves on the thickenings of its central fiber. This is a fundamental result with important applications to moduli theory. We will discuss generalizations of this result to algebraic stacks beginning with a review of the characteristic 0 situation where a satisfactory answer is known: any quotient stack $[{\rm Spec} A/G]$ whose invariant ring $A^G$ is a complete local k-algebra is coherently complete along its unique closed point. We will report on partial progress in joint work with Hall and Lim on extending this result to positive characteristic.