Motives in Montpellier

Program

There will be three mini-courses by:

Giuseppe Ancona (Université de Strasbourg) (notes)

Title: Pure motives and algebraic cycles
Abstract:

In this course I will present the category of pure motives (i.e. for smooth projective varieties) as defined by Grothendieck. Some connections with the modern category of mixed motives will be made. I will spend a little time on generalities on the category and most of the time on applications to concrete questions, especially on algebraic cycles.
A preliminary list of results we will treat is the following:

l-independency of l-adic cohomology (Katz-Messing 73),
rational and numerical equivalence coincide for elliptic curves over finite fields (Kahn 2002 and Jannsen 2007),
Bloch conjecture for some surfaces (Kimura 2005),
signature of the intersection product on abelian fourfolds (Ancona 2021).

I will insist on the different phenomena between motives over the complex numbers and motives over finite fields.

Margaret Bilu (CNRS and École polytechnique) (notes)

Title:Grothendieck rings of varieties
Abstract:

This is an introductory course on the Grothendieck ring of varieties, which has been playing an increasingly important role in algebraic geometry in the last few decades, especially with the introduction of motivic integration by Kontsevich in the mid-90s. I will present the notion together with a few useful variants, and I will construct some of the usual motivic measures, showing how these allow to recover geometric information. Then I will explain the connections with birational geometry, in particular Bittner’s presentation by blow-up relations, and the theorem of Larsen and Lunts. Some time will be spent on Kapranov’s zeta function and questions around its (non) rationality. Time permitting, I will also discuss various motivic stabilisation results in this setting.

Martin Gallauer (University of Warwick) (notes)

Title: Stable A^1-homotopy theory
Abstract:

If motives are a (universal) cohomology theory for algebraic varieties and if, instead of individual cohomology groups H^n(X), one focuses on the total cohomology H^*(X), then motives naturally become derived or homotopical objects. This "derived" or "homotopical" turn can be dated back to the 1980s and 90s, following a suggestion by Beilinson and much effort by Voevodsky, Morel, and collaborators. The goals of this mini-course are to:

explain what such a turn entails and how one can still work effectively with these things;
introduce the stable A^1-homotopy theory and highlight some of its internal structure;
describe the relation with alternative motivic theories, including those from the other two mini-courses;
touch on a few of the many applications of this viewpoint.

The winter school will also feature talks from participants.

When

The school will start on the morning of the 19th and end on the 23rd around noon.

Where

Institut Montpelliérain Alexander Grothendieck
Université de Montpellier, Batiment 10
Place Eugène Bataillon
34090 Montpellier, France

Contributed speakers

Here is the list of speakers:

Thomas AGUGLIARO (Université de Strasbourg)
Marco ARTUSA (Université de Strasbourg)
Louisa BRÖRING (Universität Duisburg-Essen)
Tom BUREL (Université Paris Cité, IMJ-PRG)
Andrea GALLESE (Scuola Normale Superiore, Pisa)
Arnab KUNDU (Simion Stoilow Institute of Mathematics of the Romanian Academy)
Elsa MANEVAL (École Polytechnique Fédérale de Lausanne)
Kenza MEMLOUK (Université de Strasbourg)
Fraser SPARKS (University of Nottingham)
Swann TUBACH (Sorbonne Université, IMJ-PRG)
Anna VIERGEVER (Leibniz Universität Hannover)

Participants

Click here for the list of participants.

Schedule

Click here to access to the schedule.

Talks

Thomas AGUGLIARO
Title: Hodge standard conjecture for powers
Abstract:

The Hodge standard conjecture predicts positivity of intersection forms on algebraic cycles. It was formulated by Grothendieck in the Sixties, motivated by an intersection theoretic proof of the Weil bound for curves over finite fields. Only recently some progress has been made, based on p-adic Hodge theory. As most conjectures on algebraic cycles, it behaves badly under powers. In this talk, we will investigate this question and prove the conjecture for powers of abelian varieties of dimension 3.

Marco ARTUSA
Title: Generalising the local Tate duality via Condensed Mathematics and the Weil group
Abstract:

Duality theorems are among the central results in arithmetic geometry. For p-adic fields, the earliest examples are due to Tate, dealing with Galois cohomology of abelian varieties and finite abelian groups. To extend this result to more general coefficients, one is forced to modify the original cohomology groups. This underlines some shortcomings of Galois cohomology, such as the lack of a natural topology on cohomology groups. In this talk, we build a new topological cohomology theory for p-adic fields, thanks to the Weil group and Condensed Mathematics. Moreover, we see how to use this cohomology theory to extend Tate’s result to more general coefficients. This new duality takes the form of a Pontryagin duality between locally compact abelian groups.

Louisa BRÖRING
Title: The 𝔸¹-Euler Characteristic of Symmetric Powers
Abstract:
The 𝔸¹-Euler characteristic of a smooth, projective scheme over a field of characteristic not two is an algebro-geometric analogue of the topological Euler characteristic. More precisely, it is a motivic measure valued not in the integers, but rather in (virtual) quadratic forms, and it carries a lot of information. For example, for a smooth, projective scheme X over ℝ, the rank of the 𝔸¹-Euler characteristic is the topological Euler characteristic of X(ℂ) and its signature is the topological Euler characteristic of X(ℝ). In general, 𝔸¹-Euler characteristics are hard to compute; for example, not much is known about its behaviour under taking quotients. In this talk, we give a gentle introduction to this invariant and we provide an overview on what is known about the 𝔸¹-Euler characteristic of the symmetric powers of a quasi-projective scheme. We then illustrate how one can use the Grothendieck-Ring of varieties to compute the 𝔸¹-Euler characteristic of low symmetric powers and symmetric powers of split toric varieties. With this, we partially confirm a conjecture of Pajwani-Pál about the 𝔸¹-Euler characteristic of symmetric powers.

Tom BUREL
Title: Equidistribution of curves in hypersurfaces
Abstract:

Let C be a curve and V a hypersurface, both complex and smooth. I will present a study of the moduli spaces Hom^e(C,V|W) of degree e morphisms from C to V verifying jet conditions W, and explain that they satisfy equidistribution in the sense of Faisant. More precisely, using a motivic framework (Grothendieck ring of varieties, motivic Euler products) this consists in obtaining a certain "quantitative" expression of said moduli spaces, asymptotically in the degree. The computations rely on the Hardy-Littlewood circle method from number theory, first converted to this motivic context by Bilu and Browning.

Andrea GALLESE
Title: How to compute the connected monodromy field of a CM abelian variety
Abstract:

Let A be an abelian variety defined over a number field k. The connected monodromy field k(eA) is the minimal extension of k over which every ell-adic Galois representation attached to A has connected image. Equivalently, it is the smallest field over which all Tate classes on self-products A^r are defined. When the extension k(eA)/k(End A) has positive degree, one finds “exotic’’ Tate classes on certain powers A^r. In this talk, I will explain how to compute the connected monodromy field for the Jacobian A of a curve with complex multiplication. After computing the endomorphism ring of A, we use CM theory to describe the algebra of Tate classes on all powers of A. We make the Galois action on this algebra explicit in terms of periods -- suitable integrals of algebraic differential forms. Although periods are generally transcendental, those attached to Tate classes are algebraic, hence computing k(eA) amounts to identifying these periods as exact algebraic numbers. This can be done numerically and, in the case of Fermat curves, via explicit algebraic identities.

Arnab KUNDU
Title: How to apply Motivic Cohomology in p-adic Hodge theory?
Abstract:

Motivic cohomology is a cohomology theory that can be defined internally within Grothendieck's category of motives. Voevodsky developed this theory for smooth varieties, demonstrating its profound connections to algebraic cycles and algebraic K-theory. However, its behaviour in mixed-characteristic remains less well understood. Building on recent advancements by Bachmann, Elmanto, Morrow, and Bouis, in a joint work with Bouis, we demonstrate a purity result over deeply-ramified bases in mixed-characteristic. I will discuss an application of this result in p-adic Hodge theory.

Elsa MANEVAL
Title: Motivic BPS invariants as motivic integrals ?
Abstract:

I will report on a work in progress, hoping to relate motivic BPS invariants to motivic integrals. I will first introduce motivic BPS invariants on one side, and give basic ideas about motivic and p-adic integration theory on the other side. In particular, I will mention the existing connexions between those integrals and BPS sheaves. Then, I will state a formula I proved, relating the motivic BPS invariant to cyclotomic inertia stacks in the Grothendieck group. It is analogous to a formula obtained by Groechenig-Wyss-Ziegler over finite fields. Then, I will explain what I hope to do next, which is to relate this further to certain motivic integrals, just as it was done for p-adic integrals.

Kenza MEMLOUK
Title: The motivic Galois group for a double zeta value
Abstract:

In this talk, we consider multiple zeta values, which are periods of unramified mixed Tate motives. For a given multiple zeta value ζ, there exists a unique minimal motive so that ζ is a period of this motive. In general, this motive is very difficult to compute. In the specific case of double zeta values, we can compute such a minimal motive. We will give the Tannakian group associated to it and discuss its dimension.

Fraser SPARKS
Title: Tt-geometry of isotropic motives
Abstract:

Tt-geometry—short for tensor-triangular geometry—is the study of classification problems of tensor-triangulated (or ‘tt-’)categories through studying associated geometric spaces: their Balmer spectra. It is a large and ongoing problem to determine the spectrum of Voevodsky’s category of (geometric) motives DM_{gm}(k) over a given field k, and the largest known collections of points come from isotropic motives (due to Vishik). In this talk I will introduce the relevant tensor-triangular background as well as isotropic motives, and talk about my work on the tt-geometry of the latter, focusing on the subcategories of isotropic (Artin—)Tate motives.

Swann TUBACH
Title: Pro-étale motives and rigidity
Abstract:

A now classical result in the theory of motives is the rigidity theorem: there is no difference between étale motivic cohomology with torsion coefficients and étale cohomology. When categorified, it says that the category of étale motives with Z/n coefficients is equivalent to the derived category of étale sheaves of Z/n modules. Using ell-completion, this enables one to construct the ell-adic realisation functor on motives. In joint work with Raphaël Ruimy and Sebastian Wolf, we construct a coefficient system of pro-étale motives, an enlargement of étale motives, that can have coefficients in any topological ring and prove that it has the 6 operations. We also prove a rigidity result: with ell-adic coefficients, one can solidify (in a condensed sense) pro-étale motives and obtain solid sheaves on the small pro-étale site, a scheme-theoritic variant of the category introduced by Fargues and Scholze. This implies that solid sheaves on schemes form a 6-functor formalism (this is not true in the perfectoid setting of Fargues and Scholze !), and provides an enhancement of the ell-adic realisation functor which is compatible with any change of coefficients (unlike the ell-completion).

Anna VIERGEVER
Title: Computing Quadratic Donaldson-Thomas Invariants
Abstract:

(Zero-dimensional) Donaldson-Thomas-invariants "count" things like ideal sheaves of a given length which have zero-dimensional support on a smooth projective complex threefold. Maulik, Nekrasov, Okounkov and Pandharipande have proven a formula for the generating series of these Donaldson-Thomas invariants in terms of the MacMahon function in the toric case. We discuss a conjectural motivic analogue of this result for smooth projective real threefolds satisfying an orientation condition, using a motivic version of Donaldson-Thomas invariants taking values in Witt rings, which are constructed using work of Levine. We provide evidence for the conjecture coming from computations for $\mathbb^3$ and $(\mathbb^1)^3$. This talk is based on my thesis and on joint work with Marc Levine.

Lunch

The lunch will take place at the CROUS Triolet every day.

Social Dinner

The social dinner will take place at the restaurant Les Vignes on Thursday evening at 8pm.

How to access the campus

The talks will take place on the campus of the Faculté des Sciences, in room 10.01 in building number 10.
The campus is easily accessible by tram. For this, take tram line 1 (stop: Saint-Eloi or Université des Sciences et Lettres) or our brand new tram line 5 (stop: Saint-Eloi or Voie Domitienne).
You can buy tickets in stations, or use the M’Ticket TAM app
If you arrive in Montpellier by plane, there is a shuttle bus that brings you near the tram station Place de l'Europe (line 1 and 4), then you will be 10 minutes walk away from the center.
If you arrive by train, there are two train stations. The station St-Roch is in the center, and the station Sud-de-France is near the tram station "Gare Sud-de-France" (line 1 again).

Organizers

Clément Dupont (clement.dupont@umontpellier.fr)
Ulysse Mounoud (ulysse.mounoud@umontpellier.fr)
Nikola Tomić (nikola.tomic@umontpellier.fr)
Sofian Tur-Dorvault (sofian.tur-dorvault@umontpellier.fr)