Motives and Hodge Theory

Anabelian representation of the motivic Galois group

I will report on recent work concerning the action of the motivic Galois group on anabelian objects such as fundamental groups of algebraic varieties conveniently completed. I'll sketch the proof of a motivic analog of a theorem of Pop (aka., the Ihara-Matsumoto-Oda conjecture) yielding an interpretation of the motivic Galois group as the automorphism group of a large diagram of anabelian objects.

The Ceresa cycle

Let C be a curve of genus >2, embedded in its Jacobian JC. The cycle [C]-[(-1)*C] is cohomologous to zero in JC; is it algebraically equivalent to zero? The answer is negative for C general (Ceresa, 1983) and for some very particular curves, and positive (trivially) for hyperelliptic curves. I will explain an example, obtained with C. Schoen, of a non-hyperelliptic curve C for which [C]-[(-1)^*C] is torsion modulo algebraic equivalence.

Stability Conditions and Moduli

Abstract. As soon as one has a stability condition on the derived category (or an admissible subcategory) of coherent sheaves on a projective variety, it is natural to inquire about the nature of the moduli of stable objects of a fixed chern class and fixed stability condition. These moduli come with determinant line bundles that relate "wall-crossings" on the stability manifold to a minimal model program for moduli spaces. This has been studied in detail for many surfaces (and non-commutative K3 categories).

In higher dimensions the situation is more complicated, but motivated by examples of Schmitt, Xia and Rezaee, we construct a family of stability conditions on projective space that converges to Gieseker stability (in an appropriate sense) and realizes all the interesting wall-crossings. This is joint work with Matteo Altavilla, Dapeng Mu and Marin Petkovic.

Hyperbolicity of moduli spaces via Hodge theory

Many interesting moduli spaces (or rather stacks) of polarized complex algebraic varieties (including moduli of curves, polarized abelian varieties, polarized K3 surfaces, polarized Calabi-Yau varieties, etc.) support a variation of Hodge structure with a quasi-finite period map. I will discuss some implications to the geometry of these moduli stacks and their finite étale covers obtained by imposing congruence conditions.

On the ring of functions on formal-analytic schemes

We will discuss explicit algebraization statements for power series related to the recent proof of Calegari-Dimitrov-Tang of the unbounded denominators conjecture. This discussion will be used as an illustration of new methods in infinite-dimensional geometry of numbers. Joint work with Jean-Benoît Bost.

Tannakian Selmer Varieties

Chabauty’s method approaches the question of rational points on a curve X by comparing with rational points on the Jacobian, in other words the degree-0 part of CH^1(X). Minhyong Kim recognized that this method could be expressed purely in terms of Galois cohomology (specifically, Selmer groups). Kim then showed how to get finer information on the rational points of X by replacing Galois cohomology groups with non-abelian cohomology *sets*, thus replacing Selmer groups with Selmer varieties. I will discuss previous and ongoing work to more explicitly understand these Selmer varieties by considering Tannakian categories of motives and/or Galois representations.

A motivic view on genus zero string amplitudes

I will review some recent joint work with Francis Brown on the structure of genus zero amplitudes in string theory. Inspired by conjectures made by physicists, we constructed motivic versions of the amplitudes and used them to prove a relation between the amplitudes associated to open strings and those associated to closed strings. They show up as two variants of a universal object on the moduli space of genus zero curves with marked points.

Rigid systems: arithmetic aspects

I’ll report on 0-dimensionial Betti moduli, so really the bottom level of the conference!: the moduli points (which are irreducible complex local systems) underlie on the p-adic scheme crystalline representations.

This is joint work with Michael Groechenig.

A non-hypergeometric E-function

In a landmark 1929 paper, Siegel introduced the class of E-functions with the goal of generalising the transcendence theorems for the values of the exponential. E-functions are power series with algebraic coefficients subject to certain growth conditions that satisfy a linear differential equation. Besides the exponential, examples include Bessel functions and a rich family of hypergeometric series. Siegel asked whether all E-functions

are polynomial expressions in these hypergeometric series. I will explain why the answer is negative (joint work with Peter Jossen).

Curves of genus three and modular forms

Some Shimura varieties can be interpreted as moduli spaces of curves. In such cases this gives information about modular forms. We discuss a ball quotient related to moduli of curves of genus three and the associated modular forms. This is joint work with Jonas Bergstroem and Fabien Clery.

Motives of moduli spaces of Higgs bundles

In joint work with Simon Pepin Lehalleur, we show the motive of the moduli space of stable Higgs bundles with coprime invariants on a curve is generated by the motive of the curve. Our geometric strategy starts with Hitchin's scaling action on the Higgs moduli space, which we use to show this motive is pure. We use this scaling action together with

Harder--Narasimhan recursions and variation of stability for moduli stacks of chains following work of García-Prada, Heinloth and Schmitt to show the motive of the Higgs moduli space is a direct factor of the motive of a sufficiently large power of the curve. We also use this scaling action to prove a motivic non-abelian Hodge correspondence: the integral motives of the Higgs and de Rham moduli spaces are isomorphic, and thus have isomorphic Chow rings.

Low degree Hurwitz stacks in the Grothendieck ring

For 1 < d < 6, we show that the class of the Hurwitz space of smooth degree d, genus g covers of P^1 stabilizes in the Grothendieck ring of stacks as g goes to infinity. We will survey the connections between this result and related stabilizations occuring in algebraic geometry, number theory, and topology.

This is based on joint work with Ravi Vakil and Melanie Matchett Wood.

Hodge theory of the Putman-Wieland conjecture

I'll discuss ongoing joint work with Aaron Landesman on the Putman-Wieland conjecture. Let f: S'-->S be a (possibly ramified) surjective map of smooth compact orientable surfaces, ramified along a finite set D, where S has genus g>1. Then the Putman-Wieland conjecture predicts that the (virtual) action of the mapping class group of (S,D) on the homology of S' has no non-zero vectors with finite orbit. The conjecture is known to be false for g=2 due to recent work of Markovic. I'll discuss some positive results in the case where the degree of f is small relative to the genus g. More generally, if f is Galois with deck transformation group G, I'll explain how to verify the conjecture for the subspace of H_1(S') spanned by G-representations of dimension at most g. The proof is Hodge-theoretic in nature.

Sparsity of Integral Points on Moduli Spaces of Varieties

Interesting moduli spaces don't have many integral points. More precisely, if X is a variety over a number field, admitting a variation of Hodge structure whose associate period map is injective, then the number of S-integral points on X of height at most H grows more slowly than H^{\epsilon}, for any positive \epsilon. This is a sort of weak generalization of the Shafarevich conjecture; it is a consequence of a point-counting theorem of Broberg, and the largeness of the fundamental group of X. Joint with Ellenberg and Venkatesh.

https://arxiv.org/abs/2109.01043

Moduli and periods for Calabi-Yau threefolds

The period map is a key tool for studying moduli spaces for K-trivial varieties, especially abelian varieties and K3 surfaces. In recent years, many of the results on period maps that hold for K3 surfaces have been generalized to hyper-Kaehler manifolds. In this talk, I will discuss (and speculate) on certain aspects of period maps for the next level of complexity, the Calabi-Yau threefolds.

Moduli of Galois covers of curves

This topic has a rich history that goes back well into the 19th century: Schwarz (1873), Picard (1880’s), Terada (1973) considered abelian covers of the projective line, culminating in the work of Deligne-Mostow (1983). The first nonabelian cases involved the icosahedral group: Wiman (1895) and later Edge (1981) considered genus 10 curve with icosahedral action (essentially completed by Farb, Dolgachev and the speaker) and Winger (1924) did the same for genus 6 curves. In this talk we go from the particular to the general: We begin with giving an almost complete story for Winger’s pencil (joint work with Yunpeng Zi). This is our stepping stone for stating a general result about the Jacobians of G-curves, of which we discuss some of its consequences and indicate its relevance for mapping class groups (joint work with Boggi).

Algebraic cycles on Gushel-Mukai varieties

I'll report on joint work (in progress) with Lie Fu, about algebraic cycles on Gushel-Mukai varieties. I'll give a brief overview of what is known in characteristic 0. I will spend most of the talk explaining the proof of the Tate conjecture for Gushel-Mukai sixfolds over fields of characteristic p at least 5.

Motivic mirror symmetry for Higgs bundles

The moduli spaces of Higgs bundles for the Langlands dual groups SL_n and PGL_n are related by a kind of mirror symmetry. In particular, there is an equality, conjectured by Hausel-Thaddeus and proved by Groechenig-Wyss-Ziegler, between their (orbifold) Hodge numbers. In joint work in progress with Victoria Hoskins, we show that this equality lifts to an isomorphism of motives. Our approach is based on the more recent proof by Maulik-Shen of the Hausel-Thaddeus conjecture together with our previous work on the motive of the moduli space of GL_n-Higgs bundles.