Geometry Over Non-Closed Fields (2024)

This meeting brought 22 researchers in algebraic and arithmetic geometry to Schloss Elmau for a program of 18 lectures, offering ample time for informal discussions and collaboration. A sample of talks include:

Quadric Surfaces Fibrations Over the Real Projective Line by Jean-Louis Colliot- Thélène: This focused on work with Pirutka addressing rationality questions for these fibrations, using innovative sums-of-squares techniques. One of their rationality constructions hinges on fine arithmetic properties of the intermediate Jacobian.

Equivariant Geometry of Singular Cubic Threefolds by Ivan Cheltsov: Nodal cubic threefolds are rational over algebraically closed fields; things get complicated when the individual singularities are not defined. This talk presented work with Marquand, Tschinkel and Zhang giving a comprehensive picture as the number of singularities grows.

Diophantine Properties of the Character Varieties by Hélène Esnault: This lecture highlighted work with de Jong and Groechenig on integral points of character varieties with applications to fundamental groups of smooth complex quasi-projective varieties.

Towards Birational Classification of 3-Folds Associated with Empty Lattice 4-Simplices by Victor Batyrev: Batyrev sketched the history of the classification of empty lattice simplices, with a view toward explicit enumeration of singularities and special families of toric hypersurfaces.

Resonance and Koszul Modules in Algebraic Geometry by Gavril Farkas: This lecture offered a unified perspective on algebra relevant to Greens conjecture and Chen invariants of hyperplane arrangements.

This list only scratches the surface of the incredible range of major results presented during the conference.

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
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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
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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
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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
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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
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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
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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
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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
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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.