MPS Conference on Arithmetic Geometry, Group Actions and Rationality Problems

Arend Bayer
University of Edinburgh

Non-Commutative Abelian Surfaces and Generalized Kummer Varieties

Polarized abelian surfaces vary in three-dimensional families. In contrast, the derived category of an abelian surface A has a six-dimensional space of deformations; moreover, based on general principles, one should expect to get “algebraic families” of their categories over four-dimensional bases. Generalized Kummer varieties (GKV) are hyperkähler varieties arising from moduli spaces of stable sheaves on abelian surfaces. Polarizd GKVs have four-dimensional moduli spaces yet arise from moduli spaces of stable sheaves on abelian surfaces only over three-dimensional subvarieties.

Arend Bayer will present a construction that addresses both issues: four-dimensional families of categories that are deformations of D^b(A) over an algebraic space. Moreover, each category admits a Bridgeland stability conditions, and from the associated moduli spaces of stable objects, one can obtain every general polarizd GKV, for every possible polariztion type of GKVs. Our categories are obtained from Z/2-actions on derived categories of K3 surfaces. This is based on joint work with Laura Pertusi, Alex Perry and Xiaolei Zhao.

Antoine Chambert-Loir
Université Paris Cité

Potential and Rationality

An 1893 theorem by Émile Borel asserts that the power series with integral coefficients that define a meromorphic function on a disk of radius > 1 is the Taylor expansion of a rational function. It has been extended in various directions (Pólya, Dwork, Bertrandias and Robinson) to encompass more complicated shapes than open disks, number fields and several absolute values. We will extend to algebraic curves of arbitrary genus the theorem of Cantor that considers Taylor expansions “at several points.” Our proof runs in two steps. The first step is an algebraicity criterion, which is proved using a method of diophantine approximation. The second step relies on the Hodge index theorem in Arakelov geometry, following an earlier work by Bost and the first author.

This is joint work with Camille Noûs.

Sildes (PDF)

Ulrich Derenthal
Hannover University

Counting Rational and Integral Points on Del Pezzo Surfaces

Fano varieties over number fields often contain infinitely many rational points. Their distribution is predicted precisely by Manin’s conjecture, which was refined, generalized and proved in many cases by Yuri Tschinkel and others. This talk will focus on Manin’s conjecture and its generalization to integral points in the case of del Pezzo surfaces.

Slides (PDF)

Pavel Etingof
MIT

Periodic Pencils of Flat Connections and their P-Curvature

A periodic pencil of flat connections on a smooth algebraic variety $X$ is a linear family of flat connections $\nabla(s_1,…,s_n)=d-\sum_{i=1}^r\sum_{j=1}^ns_jB_{ij}dx_i$, where $\lbrace x_i\rbrace$ are local coordinates on $X$ and $B_{ij}: X\to {\rm Mat}_N$ are matrix-valued regular functions. A pencil is periodic if it is generically invariant under the shifts $s_j\mapsto s_j+1$ up to isomorphism.

Pavel Etingof will explain that periodic pencils have many remarkable properties, and there are many interesting examples of them, e.g., Knizhnik-Zamolodchikov, Dunkl, Casimir connections and equivariant quantum connections for conical symplectic resolutions with finitely many torus fixed points. Etingof will also explain that in characteristic $p$, the $p$-curvature operators $\lbrace C_i,1\le i\le r\rbrace$ of a periodic pencil $\nabla$ are isospectral to the commuting endomorphisms $C_i^*:=\sum_{j=1}^n (s_j-s_j^p)B_{ij}^{(1)}$, where $B_{ij}^{(1)}$ is the Frobenius twist of $B_{ij}$. This allows us to compute the eigenvalues of the $p$-curvature for the above examples, and also to show that a periodic pencil of connections always has regular singularites. This is joint work with Alexander Varchenko.

Slides (PDF)

Julia Hartmann
University of Pennsylvania

Local-Global Principles for Tori Over Arithmetic Function Fields

Given a field $F$ and a collection of overfields $F_i$ ($I\in I$), we say that the local global principle holds for an $F$-variety $Z$ if the existence of a rational point over each $F_i$ implies the existence of an $F$-rational point.

In this talk, Julia Hartmann will study this question when $F$ is a semi-global field, i.e., the function field of a curve $X$ over a complete discretely valued field, and $Z$ is a principal homogeneous space under a torus. It is known that a local-global principle need not hold in general in this case. We study the obstruction set using R-equivalence classes and give conditions for when the local-global principle holds.

This is recent joint work with J.L. Colliot-Thélèlne, D. Harbater, D. Krashen, R. Parimala and V. Suresh.

Ljudmila Kamenova
Stony Brook University

Entire Curves on Holomorphic Symplectic Varieties

Any holomorphic symplectic manifold contains entire curves as shown by Verbitsky using ergodicity, i.e., holomorphic symplectic manifolds are non-hyperbolic. More generally, together with S. Lu and Verbitsky (and later with C. Lehn), we have established the Kobayashi conjectures in cases of Lagrangian fibrations.

In this talk, Ljudmila Kamenova will explore generalizations of these results to primitive symplectic varieties. Together with C. Lehn, we prove that if a primitive symplectic variety with second Betti number $b_2 \geq 5$ satisfies the rational SYZ conjecture, then it is hon-hyperbolic, and if $b_2 \geq 7$, then the Kobayashi pseudometric vanishes identically. This applies to all known examples of holomorphic symplectic manifolds. For Lagrangian fibrations with no multiple fibers in codimension one, we also have holomorphic dominability results with S. Lu, that imply the existence of a Zariski dense entire curve on a holomorphic symplectic manifold admitting such a Lagrangian fibration.

Slides (PDF)

Maxim Kontsevich
IHES

Atoms From GW Invariants and New Non-Rationality Results

In this talk, Maxim Kontsevich will describe applications of theory of Gromov-Witten invariants to questions of rationality. Atoms are defined in a somewhat abstract way, using eigenvalues of the operator of quantum multiplication by Euler vector field, and the blow-up formula proven by H.Iritani last year. “Bad” atoms for a given variety can be used to establish its non-rationality. I’ll explain how to work with atoms in several examples, using only the minimal information about GW invariants. The theory is especially powerful in low dimensions (3 and 4), and for non-algebraically closed fields.

This is based on a recent work in progress with L.Katzarkov, T.Pantev and T.Yu.

Bjorn Poonen
MIT

Integral Points on Curves via Baker’s Method and Finite Étale Covers

Bjorn Poonen will prove that for each g at least 2, there is no universal construction combining Baker’s method with finite étale covers to determine the integral points on all affine curves of genus g. This is joint work with Aaron Landesman.

Will Sawin
Princeton University

Reconstruction of Unitary Local Systems from Their Pushforwards

One method to construct local systems on the moduli space M_{g,n} of genus g curves with n marked points is as derived pushforwards from (ideally, simpler) local systems on M_{g,n+1}. Given the pushforward (say, as a variation of Hodge structures), can we recover the original local system?

In joint work with Daniel Litt and Aaron Landesman, Will Sawin will give a positive answer to this question under the conditions that (1) the monodromy of the original local system is unitary and (2) g is sufficiently large. This has applications to computing the monodromy of such pushforwards.

Ramin Takloo-Bighash
University of Illinois at Chicago

Rational Points and Automorphic Forms

In this talk, Ramin Takloo-Bighash will survey some old and new results on the distribution of rational and (semi) integral points on compactifications of algebraic groups using spectral methods.

The results explained in the talk are joint with Dylon Chow, Arda Demirhan, Daniel Loughran, Joseph Shalika, Sho Tanimoto and Yuri Tschinkel.

Sho Tanimoto
Nagoya University

Campana rationally connectedness and weak approximation

Campana and Abramovich introduced the notion of Campana points which interpolate between rational points and integral points. Recently there are extensive activities on arithmetic geometry of Campana points and many conjectures have been proposed. In this talk we discuss Campana curves/sections in the geometric setting. We conjecture that any Fano orbifold is Campana rationally connected. Campana conjectured that any klt Fano orbifold is Campana rationally connected. Assuming this conjecture, we prove that weak approximation at good places holds in the setting of Campana sections. This is a conjectural generalization of a theorem by Hassett and Tschinkel. Key tools to this theorem are log geometry and the notion of moduli stack of stable log maps. Finally we verify our conjecture for certain classes of orbifolds. This is work in progress which is joint with Qile Chen and Brian Lehmann.

Emmanuel Ullmo
Institut des Hautes Études Scientifiques

Bi-\Overline{Q}-Structure on Hermitian Symmetric Spaces and Quadratic Relations between CM Periods

Emmanuel Ullmo will define a natural bi-\overline{Q}-structure on the tangent space at a CM point on a Hermitian locally symmetric domain. We prove that this bi-\overline{Q}-structure decomposes into the direct sum of 1-dimensional bi-\overline{Q}-subspaces, and make this decomposition explicit for the moduli space of abelian varieties A_g. Ullmo will propose an analytic subspace conjecture, which is the analogue of the Wüstholz’s analytic subgroup theorem in this context. Ullmo will show that this conjecture, applied to A_g, implies that all quadratic Q-relations among the holomorphic periods of CM abelian varieties arise from elementary ones. Ullmo will illustrate the theory by the study of quadratic relations between CM periods of CM abelian varieties of anti-Weyl type.

This is joint work with Z. Gao and A. Yafaev.

Slides (PDF)

János Kollár

Essential Dimension of Isogenies

Let A be an abelian variety of dimension g. János Kollár will show that the essential dimension of the multiplication by m isogeny of A equals g, confirming a conjecture of Brosnan. This is joint work with Z. Zhuang.