MPS Conference on Arithmetic Geometry, Group Actions and Rationality Problems

Date & Time

This meeting, held at the Simons Foundation Fischbach Auditorium, brought about 40 researchers from around the world to present recent research results and survey the development of various fields. Notable talks included:

  • “Non-Commutative Abelian Surfaces and Generalized Kummer Varieties” by Arend Bayer: An important contribution to the geometry of hyperkähler manifolds of the Kummer type, interpreting them as moduli spaces of objects over non-commutative deformations of abelian surfaces and offering a new approach to the Kuga-Satake construction;
  • “Potential and Rationality” by Antoine Chambert-Loir: The point of departure was an 1893 result of Émile Borel characterizing series expansions of rational functions, interpreted as meromorphic functions on the Riemann sphere. Here the focus was on higher-genus curves, with technical inputs from Arakelov geometry and potential theory;
  • “Counting Rational and Integral Points on Del Pezzo Surfaces” by Ulrich Derenthal: a remarkable survey of two decades of work inspired by Manin’s conjecture and its refinements;
  • “Atoms from GW Invariants and New Non-Rationality Results” by Maxim Kontsevich: A revolutionary strategy for establishing non-rationality of the general complex cubic fourfold, with implications for rationality problems over non-closed fields;
  • “Essential Dimension of Isogenies” by János Kollár: This established the incompressibility of multiplication maps on abelian varieties;
  • “Integral Points on Curves via Baker’s Method and Finite Étale Covers” by Bjorn Poonen: Can integral points on hyperbolic affine curves be understood completely via linear forms in logarithms, applied to étale covers? Poonen showed this cannot work, at least following the most straightforward formulation.

The workshop included a reception where students, collaborators and colleagues of Yuri Tschinkel spoke briefly about his influence, both mathematical and personal, over many years.

Brendan Hassett, Brown University
Ludmil Katzarkov, University of Miami
Tony Pantev, University of Pennsylvania

  • Agendaplus--large

    Wednesday, May 29

    9:30 AMMaxim Kontsevich | Atoms from GW Invariants and New Non-Rationality Results
    10:30 AMBREAK
    11:00 AMArend Bayer | Non-Commutative Abelian Surfaces and Generalised Kummer Varieties
    12:00 PMLUNCH
    1:00 PMWill Sawin | Reconstruction of Unitary Local Systems from their Pushforwards
    2:00 PMBREAK
    2:30 PMJulia Hartmann | Local-Global Principles for Tori Over Arithmetic Function Fields
    3:30 PMBREAK
    4:00 PMBjorn Poonen | Integral Points on Curves via Baker's Method and Finite Étale Covers

    Thursday, May 30

    9:30 AMJános Kollár | Essential Dimension of Isogenies
    10:30 AMBREAK
    11:00 AMUlrich Derenthal | Counting Rational and Integral Points on Del Pezzo Surfaces
    12:00 PMLUNCH
    1:00 PMLjudmila Kamenova | Entire Curves on Holomorphic Symplectic Varieties
    2:00 PMBREAK
    2:30 PMAntoine Chambert-Loir | Potential and Rationality
    3:30 PMBREAK
    4:00 PMSho Tanimoto | Campana Rationally Connectedness and Weak Approximation

    Friday, May 31

    9:30 AMPavel Etingof | Periodic Pencils of Flat Connections and their P-Curvature
    10:30 AMBREAK
    11:00 AMRamin Takloo-Bighash | Rational Points and Automorphic Forms
    12:00 PMLUNCH
    1:00 PMEmmanuel Ullmo | Bi-\Overline{Q}-Structure on Hermitian Symmetric Spaces and Quadratic Relations between CM Periods
  • Abstracts & Slidesplus--large

    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

    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

    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

    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.

  • Contactsplus--large

    Registration and Travel Assistance
    Ovation Travel Group
    [email protected]
    (917) 408-8384 (24-Hours)

    Meeting Questions and Assistance
    Meghan Fazzi
    Manager, Events and Administration, MPS, Simons Foundation
    [email protected]
    (212) 524-6080


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