# Amherst 2022

AGNES Fall 2022 was held at UMass Amherst from Friday November 18 to Sunday November 20, 2022.

**Videos of talks**

**Juliette Bruce** (Brown)

*The top-weight cohomology of A_g. ** *talk

I will discuss recent work calculating the top weight cohomology of the moduli space A_g of principally polarized abelian varieties of dimension g for small values of g. The key idea is that this piece of cohomology is encoded combinatorially via the relationship between the boundary complex of a compactification of A_g and the moduli space of tropical abelian varieties. This is joint work with Madeline Brandt, Melody Chan, Margarida Melo, Gwyneth Moreland, and Corey Wolfe.

**Mark Andrea de Cataldo**** **(Stony Brook)

*Cohomological Non Abelian Hodge Theorem for curves in positive characteristic. *pre-talk , talk

The Non Abelian Hodge Theory (NAHT) of Simpson, Corlette, et al. yields canonical diffeomorphisms between the moduli spaces of Higgs bundles, flat connections, and representations of the fundamental group of a curve. In positive characteristic, there is a ``twisted" version of NAHT, but it is not clear how to extract geometric information from it.

I will report on joint work with graduating student Siqing Zhang, where we prove a cohomological version of NAHT, i.e., we exhibit a canonical isomorphism between the etale cohomology rings of the moduli of Higgs bundles and of flat connections.

I will explain how this works via vanishing cycle theory.

TIme permitting, I will also discuss some perhaps unexpected corollaries relating cohomology rings of different moduli spaces, in equal and in mixed characteristic.

**Sarah Frei**** **(Dartmouth)

*Arithmetic and derived categories. *talk

The derived category of coherent sheaves has proven to be a useful tool for studying the geometry of algebraic varieties. In this talk, I will discuss various ways in which the derived category does and doesn't capture the arithmetic of algebraic varieties. I will focus on the existence of rational points and cohomology as Galois representations. Along the way, I'll point out interesting open questions in this area. I will end with an example in which, conversely, understanding the arithmetic of a variety can tell us about derived equivalences.

**Daniel Halpern-Leistner**** **(Cornell)

*The noncommutative minimal model program. *talk

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.

**Alexander Perry** (Michigan)

*The period-index problem. *pretalk* , *talk

In the theory of Brauer groups, the longstanding period-index problem asks for a bound on one measure of complexity of a central simple algebra (its index) in terms of another (its period). I will discuss recent progress on this problem which relies on a mixture of ideas from Hodge theory, noncommutative/derived algebraic geometry, and enumerative geometry. This is based on joint works with James Hotchkiss and with Johan de Jong.

**Mihnea Popa**** **(Harvard)

*Conjectures on the Kodaira dimension of algebraic varieties. *pretalk , talk

* *Algebraic varieties are classified in terms of various numerical invariants, among which one of the most basic is the Kodaira dimension. A celebrated conjecture of Iitaka, that has driven many developments in birational geometry, predicts that any morphism of smooth projective varieties exhibits a certain subadditive behavior for the Kodaira dimension of its domain, target and general fiber. I will propose some superadditivity conjectures complementing it, and describe a few cases where progress has been made.

**Julie Rana** (Lawrence)

*Standard stable Horikawa surfaces. *talk

Smooth surfaces with invariants on the Noether line were completely described by Horikawa and so are known as Horikawa surfaces. When the volume K^2 is a multiple of 8, the moduli space of these surfaces consists of two disconnected components. In joint work with S. Rollenske, we show that the closures of these two components intersect in a divisor parametrizing explicitly described semi-smooth surfaces. Moreover, for any value of K^2 along the Noether line, our methods give rise to new, generically non-reduced, irreducible components of increasing dimensions in the same connected component of the classical moduli space of Horikawa surfaces.

**Jakub Witaszek**** **(Princeton)

*Classification of algebraic varieties in mixed characteristic. *talk (Note: unfortunately only the last 1/3 was recorded.)

In my talk I will describe how recent breakthroughs in arithmetic geometry and commutative algebra led to new developments in classifying algebraic varieties in mixed characteristic. This is based on joint work with Bhargav Bhatt, Linquan Ma, Zsolt Patakfalvi, Karl Schwede, Kevin Tucker, and Joe Waldron, as well as with Christopher Hacon.

This conference was made possible by support from the National Science Foundation under grant numbers DMS-1937705 (University of Massachusetts Amherst) and DMS-1937370 (Rutgers University).

Any opinions, findings and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation (NSF).