Geometry Over Non-Closed Fields (2024)

*Participants may explore the hotel property and its surrounding areas as well as engage in informal discussion with other participants.

Benjamin Bakker
University of Illinois at Chicago

Integral Canonical Models and Period Maps

The work of Chai–Faltings, Milne, Moonen, Kisin and Kottwitz constructs integral canonical models for abelian type Shimura varieties, but as the construction relies on the modular interpretation of the moduli space of abelian varieties, it does not apply to exceptional Shimura varieties. In joint work with A. Shankar and J. Tsimerman, we construct integral canonical models for all Shimura varieties at sufficiently large primes, as well as for the image of any period map arising from geometry. Our method passes through finite characteristic and relies on a partial generalization of the work of Ogus–Vologodsky. As applications, in the context of exceptional Shimura varieties, we prove analogs of Tate semisimplicity in finite characteristic, CM lifting theorems for ordinary points and the Tate isogeny theorem for ordinary points.
 

Victor Batyrev
Universität Tübingen

Towards Birational Classification of \(3\)-Folds Associated with Empty Lattice 4-Simplices

A lattice \(d\)-simplex \(\Delta \subset {\mathbb R}^d\) is called {\em empty} if it has no integer points other than its vertices. There exists a natural bijection between empty lattice \(d\)-simplices \(\Delta\) and \((d+1)\)-dimensional Gorenstein terminal affine simplicial toric varieties. In case \(d=3\), the complete classification of \(4\)-dimensional Gorenstein terminal affine simplicial toric varieties was obtained by D. Morrison and J. Stevens.

The present talk applies the mirror duality to \(\Delta\). We regard \(\Delta\) as Newton polytope of an associated hypersurface \(Z_\Delta\subset {\mathbb G}_m^d\) over an arbitrary field \(K\). Note that the geometric genus of \(Z_\Delta\) is always zero. We study the birational properties of \(Z_\Delta\) over \(K\). If \(d =3\), by a combinatorial theorem of G.

White, the surface \(Z_\Delta \subset {\mathbb G}_m^3\) is always \(K\)-rational. However, if \(d=4\), this is no longer the case. We use the complete unimodular classification of empty lattice \(4\)-simplices, which was initiated by S. Mori, D. Morrison and I. Morrison in the late eighties and completely finished recently by {\’O}.

Iglesias-Valiño and F. Santos, to show that among all infinitely many empty lattice \(4\)-simiplices \(\Delta\) there exist exactly \(552\) ones such that the Kodaira dimension of \(Z_\Delta\) is non-negative. Using a recently developed general method, we are able to find minimal models of \(Z_\Delta\) under condition that the characteristic of \(K\) does not divide the lattice normalized volume of \(\Delta\).

This is joint work with Martin Bohnert.
 

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. Polarized GKVs have four-dimensional moduli spaces yet arise from moduli spaces of stable sheaves on abelian surfaces only over three-dimensional subvarieties.

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 condition, and from the associated moduli spaces of stable objects, one can obtain every general polarized GKV, for every possible polarization type of GKVs. Our categories are obtained from Z/2-actions on derived categories of K3 surfaces.

This lecture will focus on Weil-type abelian fourfolds and connections to Kuga-Satake constructions obtained by Markman and O’Grady.

This is based on joint work with Laura Pertusi, Alex Perry and Xiaolei Zhao.
 

Christian Böhning
University of Warwick

Equivariant Birational Types and Derived Categories

In this talk, we will investigate equivariant birational geometry of rational surfaces and threefolds from the perspective of derived categories. We will also discuss G-Pfaffian cubic fourfolds. It will be shown that the natural analogues of conjectures of Lunts and Kuznetsov, relating structures in derived categories with rationality, do not hold in the equivariant context. This is joint work with Hans-Christian von Bothmer, Yuri Tschinkel and Brendan Hassett.
 

Anna Cadoret
IMJ-PRG, Sorbonne Université

Bounding Uniformly in Families the Obstruction to the Hodge and Tate Conjectures

It is well known that the integral variants of the Hodge and Tate conjectures may fail even if their rational (classical) variants hold. Assuming the rational variants of these conjectures, the obstruction to the integral variants is measured by a certain finite group. We explain how to bound uniformly this group in families of algebraic varieties under suitable genericity assumptions. As a corollary, we obtain a uniform bound of the obstruction for families of varieties parametrized by the k-rational points (and even the points of bounded degree) of a curve over a finitely generated field k. This is a joint work with Alena Pirutka.
 

Antoine Chambert-Loir
Université Paris Cité

Burnside Rings and Volume Forms with Logarithmic Poles

The Burnside rings of varieties is the free abelian group on birational classes of varieties with a natural ring structure induced by product of varieties. The proof by Kontsevich and Tschinkel that rationality specializes in smooth families is built on a specialization morphism for these rings. It also led to a rich study of invariants of birational morphisms, in particular by Lin and Shinder, as well as an equivariant generalization by Kontsevich and Kresch.

We will discuss another generalization of this theory in the context of algebraic varieties equipped with logarithmic volume forms. We introduce a residue homomorphism and construct an additive invariant of birational morphisms. We also define a specialization homomorphism.

This is joint work with Maxim Kontsevich and Yuri Tschinkel.
 

Ivan Cheltsov
University of Edinburgh

Equivariant Geometry of Singular Cubic Threefolds

In this talk, we will consider the geometric counterpart of the following question: Which singular cubic threefolds are rational over algebraically non-closed field? Namely, we will discuss the linearizability of actions of finite groups on singular cubic threefolds, using cohomological tools, intermediate Jacobians, Burnside invariants, and the equivariant Minimal Model Program. This is a joint work with Lisa Marquand, Yuri Tschinkel and Zhijia Zhang.
 

Jean-Louis Colliot-Thélène
CNRS et Université Paris-Saclay

Quadric Surfaces Fibrations Over the Real Projective Line

We consider the question whether a real threefold fibred into quadric surfaces over the real projective line is stably rational (over the reals) if the topological space of real points is connected. We give a counterexample. When all geometric fibres are irreducible, the question is open. We investigate a family of such fibrations for which the intermediate Jacobian technique is not available. For these, we produce two independent methods which in many cases enable one to prove decomposition of the diagonal.

This is joint work with Alena Pirutka.
 

Ulrich Derenthal
Leibniz Universität Hannover

Manin’s Conjecture for Spherical Fano Threefolds

For Fano varieties over number fields, the asymptotic behavior of the number of rational points of bounded height is predicted by Manin’s conjecture. We discuss a proof of Manin’s conjecture for smooth spherical Fano threefolds. In one case, in order to obtain the expected asymptotic formula, it is necessary to exclude a thin subset with exceptionally many rational points from the count. This is joint work with V. Blomer, J. Brüdern and G. Gagliardi.
 

Hélène Esnault
FU Berlin / Harvard / Copenhagen

Diophantine Properties of the Character Varieties

We show (as an example) that the character variety of a smooth complex quasi-projective variety, if non-empty and geometrically irreducible, possesses an *integral* point. More generally, if non-empty, it possesses \(\bar{\mathbb Z}_\ell\) points for all \(\ell\).

The method, based on the arithmetic Langlands program (via companions) and the geometric Langlands program (via de Jong’s conjecture) produces a new obstruction for a finitely presented group to be the fundamental group of a smooth complex quasi-projective variety.

This is joint work with Johan de Jong, partly based on work with Michael Groechenig.
 

Gavril Farkas
Humboldt Universität zu Berlin

Resonance and Koszul Modules in Algebraic Geometry

Inspired from ideas in topology, Koszul modules and the associated resonance varieties turned out to have important algebro-geometric applications for instance to (i) Green’s Conjecture on syzygies of canonical curves, (ii) stabilization of cohomology of projective varieties in arbitrary characteristics and (iii) Chen invariants of hyperplane arrangements. I will discuss the latest developments related to this circle of ideas obtained in joint work with Aprodu, Raicu and Suciu.
 

Klaus Hulek
Leibniz University Hannover

Ball Quotients and Moduli Spaces

A number of moduli problems are, via Hodge theory, closely related to ball quotients. In this situation, there is often a choice of possible compactifications such as the GIT compactification´and its Kirwan blow-up or the Baily–Borel compactification and the toroidal compactification. The relationship between these compactifications is subtle and often geometrically interesting. In this talk, I will discuss several cases, including cubic surfaces and threefolds and Deligne-Mostow varieties. This discussion links several areas such as birational geometry, moduli spaces of pointed curves, modular forms and derived geometry. This talk is based on joint work with S. Casalaina-Martin, S. Grushevsky, S. Kondo, R. Laza and Y. Maeda.
 

Ludmil Katzarkov
University of Miami

Atoms, Equivariant Atoms and Applications

In this talk, we will introduce the theory of atoms — new birational invariants. Many examples will be considered.
 

Edgar Jose Martins Dias Costa
MIT

Effective Computation of Hodge Cycles

In this talk, we will overview several techniques to compute the Hodge cycles on surfaces. We will start by studying the self-product of a curve, with the goal of computing the endomorphism of its Jacobian. We will follow this with the problem of computing the Picard lattice of a K3 surface.
 

Alexander Schmidt
Universität Heidelberg

Generalized Anabelian Geometry

Grothendieck’s anabelian philosophy predicts the existence of a class of anabelian varieties X that are reconstructible from their étale fundamental group. All examples of anabelian varieties known so far are of type K(\pi ; 1), i.e., their higher étale homotopy groups vanish. For general varieties X, the homotopy theoretic viewpoint suggests to ask the modified question, whether they are reconstructible from their étale homotopy type instead of only the fundamental group.

In the talk, we present some progress in direction of this “generalized anabelian geometry.”
 

Alexei Skorobogatov
Imperial College London

Hasse Principle for Intersections of Two Quadrics via Kummer Surfaces

Smooth intersections of two quadrics in the projective space of dimension at least 5 over a number field are expected to satisfy the Hasse principle. This was proved by Wittenberg in his thesis, conditionally on the finiteness of the Tate–Shafarevich groups of elliptic curves and Schinzel’s Hypothesis. In a joint work with Adam Morgan, we remove the dependence on Schinzel’s Hypothesis, while assuming the finiteness of the Tate–Shafarevich groups of Jacobians of genus 2 curves. The proof proceeds by proving the Hasse principle for Kummer surfaces attached to 2-coverings of such Jacobians satisfying certain local conditions and deducing the Hasse principle for smooth intersections of two quadrics in the projective space of dimension 4 with irreducible characteristic polynomial, which is known to imply the Hasse principle for intersections of quadrics in higher dimension. We also prove a similar result when the characteristic polynomial is completely split.
 

Andras Szenes
University of Geneva

Intersection Cohomologies of Moduli Spaces of Vector Bundles on Curves

The study of the intersection cohomologies of the moduli spaces of semistable bundles on Riemann surfaces goes back to the early works by Frances Kirwan in the ‘80s, and has been the focus of a lot of research ever since. In joint work with Camilla Felisetti and Olga Trapeznikova, we found a simple method of calculating the intersection Betti numbers of these spaces.

The central idea is to apply a refinement of the Decomposition Theorem to the parabolic projection map. I will give a gentle introduction to the subject and then formulate the main results.
 

Sho Tanimoto
Nagoya University

The Spaces of Rational Curves on Del Pezzo Surfaces via Conic Bundles

There have been extensive activities on counting functions of rational points of bounded height on del Pezzo surfaces, and one of prominent approaches to this problem is by the usage of conic bundle structures on del Pezzo surfaces. This leads to upper and lower bounds of correct magnitude for quartic del Pezzo surfaces. In this talk, I will explain how conic bundle structures on del Pezzo surfaces induce fibration structures on the spaces of rational curves on such surfaces. Then I will explain applications of this structure which include: (1) upper bounds of correct magnitude for the counting function of rational curves on quartic del Pezzo surfaces over finite fields and (2) rationality of the space of rational curves on a quartic del Pezzo surface.

Inspired by these ideas, I will explain our ongoing proof of homological stability for the spaces of rational curves on quartic del Pezzo surfaces. The last work on homological stability is joint work in progress with Ronno Das, Brian Lehmann and Philip Tosteson.

Participants will have access to a projector and screen for computer-based talks and blackboards for those who prefer to give board-based talks.

High-speed Internet access is available.

Funding
The foundation will arrange and pay for all air and train travel to the Simons Symposium as well as hotel accommodations and reimbursement of local expenses. Business-class or premium economy airfare will be booked for all flights over five hours.

Participation
The Simons Symposium only accommodates full in-person participation. By accepting, you agree to participate fully in the scientific program, arriving on Sunday prior to the welcome dinner and departing on the following Saturday. No accommodations will be made for partial or remote attendance.

Guests & Children
It is against Simons Foundation policy for guests, including children, to accompany participants during the meeting week. Guests are welcome to join prior to or at the conclusion of the symposium at their own expense.

Air & Rail
The foundation will arrange and pay for round-trip travel from a participant’s home city to the symposium. All travel and hotel arrangements must be booked through the Simons Foundation’s preferred travel agency. Travel arrangements not booked through the preferred agency, including triangle trips and routing/preferred airlines outside budget, must be pre-approved by the Simons Foundation and a reimbursement quote must be obtained through the foundation’s travel agency.

Travel specifications, including preferred airline, will be accommodated provided that these specifications are reasonable and within budget.

All costs related to changes made to ticketed travel are to be paid for by the participant and are not reimbursable. Please contact the foundation’s travel agency for further assistance.

Personal & Rental Cars
Personal car and rental car trips over 250 miles each way require prior approval from the Simons Foundation via email.

Regardless of distance, rental cars must be pre-approved by the Simons Foundation.

The Simons Foundation will directly arrange roundtrip ground transportation between the arriving airport and the symposium hotel.

Expenses for ground transportation outside the above arrangements will be reimbursed for travel days (i.e. traveling to/from the airport) as well as local transportation.

Overview
Individuals will be reimbursed for meals and local expenses including ground transportation. Expenses should be submitted through the foundation’s online expense reimbursement platform after the meeting’s conclusion.

Expenses accrued as a result of meetings not directly related to the Simons Foundation-hosted meeting will not be reimbursed by the Simons Foundation and should be paid by other sources.

Below are key reimbursement takeaways; a full policy will be provided with the final logistics email circulated approximately 2 weeks prior to the meeting’s start.

Meals
The Simons Foundation will coordinate all meals during the symposium weeks. The daily meal limit for all other meals (i.e. those taken on travel days) is $125 and itemized receipts are required for expenses over $24 USD. The foundation DOES NOT provide a meal per diem and only reimburses actual meal expenses.

• Meals taken on travel days are reimbursable.
• Meals taken outside those provided by the foundation (breakfast, lunch, breaks and/or dinner) are not reimbursable.
• If a meal was not provided on a meeting day, dinner for example, that expense is reimbursable.
• Meals taken on days not associated with Simons Foundation-coordinated events are not reimbursable.
• Minibar expenses are not reimbursable.
• Meal expenses for a non-foundation guest are not reimbursable.

Group meals consisting of fellow meeting participants paid by a single person will be reimbursed up to $65 per person per meal and the amount will count towards each individual’s $125 daily meal limit.

Ground Transportation
Expenses for ground transportation will be reimbursed for travel days (i.e. traveling to/from the airport) as well as local transportation. While in NYC, individuals are encouraged to use public transportation and not use taxi, Uber or Lyft services.