P-adic and Characteristic p Methods in Algebraic Geometry (2-13 June 2025) ‒ CAG ‐ EPFL

This is a 2 weeks program at the Bernoulli Center about p-adic and characteristic p methods in algebraic geometry. The outline of the program is:

2 June – 6 June conference

7 June – 9 June free days, including Monday because of Pentecost, access can be granted to the Bernoulli center if needed, otherwise we can suggest plenty of tourist activities

10 June – 13 June workshop with 2 or 3 talks in the morning, then open problem session, and then group discussions on the open problems.

See detailed schedule below.

Location:

everything will happen in the the Bernoulli Center, which is on the 3rd floor (with European counting, so 4th with the US counting) of the building
for lunch, we plan with participants using any of the campus eateries or the food-trucks, or even one of the coffee shops in case of a light lunch. A map of some of the different options is:

Detailed schedule:

2 June

9:00-10:00 Christian Liedtke

Title: On Kleinian singularities

Abstract: In 1884, Felix Klein classified the finite subgroups of SL(2,C) (C=complex numbers) and computed the associated quotient singularities, which coincide with the rational double point singularities, as well as with the canonical surface singularities. In my talk, I will extend these results to finite and linearly reductive subgroup schemes of SL(2,k), where k is an arbitrary field, and even to SL(2,R), where R is the ring of integers in a number field. In the first case, this leads to singularities not only of type ADE, but also of types BCFG (the finite Dynkin diagrams that are not simply laced). In the second case, only types A and B show up and I will establish finiteness and density results using global class field theory. This is joint work (partly in progress) with Matt Satriano.

10:00-10:20 coffee break

10:20-11:20 Adrian Langer

Title: Simpson’s correspondence and intersection theory on normal varieties in positive characteristic

Abstract: The talk will be devoted to various analogues of the results of Donaldson-Uhlenbeck-Yau, Simpson and others for varieties in positive characteristic. Special attention will be given to quasi-projective varieties, where our study leads to interesting problems concerning standard notions of positivity for vector bundles. This also gives rise to challenges related to intersection theory on normal varieties.

11:20-11:40 coffee break

11:40-12:40 Kęstutis Česnavičius

Title: Generically trivial torsors under constant groups

Abstract: For a smooth variety X over a field k and a smooth k-group scheme G, Grothendieck and Serre predicted that every generically trivial G-torsor over X trivializes Zariski locally on X. In the case when G is the automorphism group of a projective k-variety, this specializes to a Zariski local triviality statement for isotrivial, generically trivial families of varieties parametrized by X. I will explain a resolution of the Grothendieck–Serre problem, the main new case being when k is imperfect, in which pseudo-reductive and quasi-reductive groups play a central role. The talk is based on joint work with Alexis Bouthier and Federico Scavia.

12:40-14:30 Lunch break

14:30-15:30 Christopher Lazda

Title: An overconvergent Riemann-Hilbert correspondence

Abstract: According to a philosophy of Grothendieck, every good cohomology theory should have a six functor formalism. Arithmetic D-modules were introduced by Berthelot to provide the theory of rigid cohomology with exactly such a formalism. However, it is not clear that cohomology groups computed via the theory of arithmetic D-modules coincide with the analogous rigid cohomology groups. In this talk I will describe an overconvergent Riemann-Hilbert correspondence that can be used to settle this question.

15:30-16:00 coffee break

16:00-17:00 Dino Lorenzini

Title: Resolutions of Wild Quotient Singularities of Surfaces in Characteristic p

Abstract: Starting with an automorphism of the power series ring A:=k[[x,y]] of prime order p, one can try to compute its invariant subring B. When the ring B is not regular, one can hope to resolve the singularity of Spec(B), and obtain from the exceptional divisor of the resolution a graph and a symmetric negative-definite integer intersection matrix describing the combinatorics of the resolution. I will survey in this talk what is known about the possible matrices that can arise in this way when p is also the characteristic of the field k, and discuss some open problems.

17:00-18:30 Happy hour

3 June

9:00-10:00 Karl Schwede

Title: Ideal closure operations in characteristic zero via resolution of singularities

Abstract: In characteristic p > 0, the origins of many classes of singularities come out of tight closure theory. Even more recent mixed characteristic singularity classes are closely related to things like epf closure. In this talk, I will explore a characteristic zero theory which is inspired by the positive and mixed characteristic theories. This theory uses resolution of singularities, Grauert-Riemenschneider vanishing and the dg-algebra structure of the derived pushforward of the structure sheaf, to establish its basic properties. As an application, we will discuss what this operation says about those classical closure operations for p >> 0. This is joint work with Neil Epstein, Peter McDonald, and Rebecca R.G.

10:00-10:20 coffee break

10:20-11:20 Linquan Ma

Title: Perfectoid pure singularities

Abstract: We introduce a mixed characteristic analog of F-pure singularities, which we call perfectoid pure singularities. We will present some basic and expected properties of these singularities including their connections with log canonical singularities. We will then show how to produce examples of these singularities via deformation to positive characteristic and discuss some related open questions. This talk is based on joint work with Bhargav Bhatt, Zsolt Patakfalvi, Karl Schwede, Kevin Tucker, Joe Waldron, and Jakub Witaszek.

11:20-11:40 coffee break

11:40-12:40 Joe Waldron

Title: Mixed characteristic analogues of Du Bois and log canonical singularities

Abstract: Singularities are measured in different ways in characteristic zero, positive characteristic, and mixed characteristic. However, classes of singularities usually form analogous groups with similar properties, with an example of such a group being klt, strongly F-regular and BCM-regular. In this talk we shall focus on newly introduced mixed characteristic counterparts of Du Bois and log canonical singularities and discuss their properties. This is joint work with Bhargav Bhatt, Linquan Ma, Zsolt Patakfalvi, Karl Schwede, Kevin Tucker and Jakub Witaszek.

12:40-14:30 Lunch break

14:30-15:30 Kevin Tucker

Title: Plus-pure thresholds of some cusp-like singularities

Abstract: The log canonical threshold (lct) is an important numerical invariant of singularities in complex algebraic geometry, with analytic origins. Via standard reduction to characteristic $p>0$ techniques, it is closely related to the F-pure threshold in positive characteristic defined in terms of the Frobenius endomorphism. These equal characteristic thresholds admit an analogue in the developing theory of singularities in mixed characteristic, which is known as the plus-pure threshold. In this talk, I will review these notions and discuss a computation of the plus-pure thresholds of some mixed characteristic cusp-like singularities (such as $p^2+x^3 \in \mathbb{Z}_p[[x]]$). This talk is based on joint work with Hanlin Cai, Suchitra Pande, Eamon Quinlan-Gallego, and Karl Schwede.

15:30-16:00 coffee break

16:00-17:00 Ryo Ishizuka

Title: A unified construction of perfectoid towers by prisms

Abstract: The theory of perfectoid towers, developed by Ishiro-Nakazato-Shimomoto, provides a systematic way to connect Noetherian rings with perfectoid rings. This theory can be seen as a tower-theoretic generalization of perfectoid rings. However, the known explicit examples of perfectoid towers are limited. On the other hand, Bhatt and Scholze introduced prisms as a fundamental object in prismatic cohomology theory. Prisms also generalize perfectoid rings. In this talk I will explain my result showing that a wide class of prisms gives rise to perfectoid towers. This leads to a more comprehensive construction of perfectoid towers and provides many new examples.

17:00-18:30 Happy hour

4 June

10:00-11:00 Jakub Witaszek

Title: Hodge Theory of Singularities in Positive Characteristic

Abstract: I will start by reviewing various Hodge-theoretic invariants of complex singularities. After that, I will discuss an approach to interpreting these invariants using methods from positive characteristic. This is based on joint work with Tatsuro Kawakami

11:00-11:30 coffee break

11:30-12:30 Tatsuro Kawakami

Title: Extending one-forms on F-regular singularities

Abstract: For a normal variety X, we say that X satisfies the logarithmic extension theorem for i-forms if, for every proper birational morphism f : Y to X, every i-form on the regular locus of X extends to a logarithmic i-form on Y. This property fundamentally relates differential forms and singularities, and many results are known over the field of complex numbers. In this talk, we discuss the extension theorem in positive characteristic. In particular, we prove the logarithmic extension theorem for one-forms on strongly F-regular singularities by utilizing Cartier operators. Joint work with Kenta Sato.

12:40-14:30 Lunch break

14:30-15:30 Jefferson Baudin

Title: A Grauert-Riemenschneider vanishing theorem for Witt canonical sheaves

Abstract: Vanishing theorems are ubiquitous in characteristic zero birational geometry. A particularly useful one is Grauert-Riemenschneider vanishing, which asserts that if f : Y -> X is a projective birational morphism and Y is smooth, then higher pushfowards of \omega_Y vanish.
On the other hand, this vanishing theorem famously fails in positive characteristic. In this talk, we will explain how to prove a Witt vector version of Grauert-Riemenchneider vanishing, solving a question of Blickle and Esnault and later studied by several authors. If time permits, we will give applications to singularities and to the cohomology of irregular ordinary varieties.

15:30-16:00 coffee break

16:00-17:00 Kenta Sato

Title: Uniform lower bound for Seshadri constants

Abstract: In 1995, Ein, Küchle, and Lazarsfeld proved that the Seshadri constant at a very general point of a nef and big line bundle on a projective variety X over a field of characteristic zero is at least the inverse of the dimension of X. However, in positive characteristic, no uniform lower bound for Seshadri constants is known, even in dimension two. In this talk, we provide a partial answer to this problem in dimension three. As an application, we give a partial affirmative result toward the BAB conjecture for threefolds in characteristic p>5.

17:00-18:30 Happy hour

5 June

9:00-10:00 Valentjin Karemaker

Title: When is a polarised abelian variety determined by its p-divisible group

Abstract: We will study the moduli space of abelian varieties in characteristic p and in particular its supersingular locus. We first determine precisely when this locus is geometrically irreducible. Since it was known that the number of components is a class number, this comes down to solving a “class number one problem” or “Gauss problem”. Next, we will show when a polarised abelian variety is determined by its p-divisible group. This can be viewed as a Gauss problem for central leaves, which are the loci consisting of points whose associated p-divisible groups are isomorphic. Our solution involves mass formulae, computations of automorphism groups, and a careful analysis of Ekedahl-Oort strata in genus 4. This is based on joint work with Ibukiyama and Yu.

10:00-10:20 coffee break

10:20-11:20 Teppei Takamatsu

Title: Quintic del Pezzo threefolds in positive and mixed characteristic

Abstract: A quintic del Pezzo threefold, or a V5-variety, which is a Fano threefold with index 2 and (-K/2)^3 =5, is one of the most fundamental, but nontrivial, Fano threefolds after the projective space P^3 or the hyperquadric Q^3. In this talk, I will introduce our work on the classification and structural study of V5-varieties over an arbitrary base scheme, and present its application to the explicit finiteness of V5-varieties over a ring of integers (so-called the Shafarevich conjecture). If time permits, I will also explain how these results apply to the study of prime Fano 3-folds of genus 12 in positive and mixed characteristic. This talk is based on joint work with Tetsushi Ito, Akihiro Kanemitsu, and Yuuji Tanaka.

11:20-11:40 coffee break

11:40-12:40 Shou Yoshikawa

Title: Quasi-F-splitting and perfectoid purity

Abstract: Quasi-F-splitting is a type of singularity in positive characteristic, introduced by Yobuko. In this talk, I will extend the notion of quasi-F-splitting to rings in mixed characteristic, and explain its relationship with perfectoid purity. Furthermore, I will introduce a method for computing perfectoid purity thresholds, along with some new examples.

12:40-14:30 Lunch break

14:30-15:30 Hanlin Cai

Title: Characterizing perfectoid covers of abelian varieties

Abstract: In this talk, I will explain how to give a simple characterization of determining whether a profinite etale cover of an abelian variety is perfectoid. I will also explain how this result is motivated by (the geometric) Sen theory and confirms a conjecture of Rodríguez Camargo on perfectoidness of p-adic Lie torsors for abelian varieties. This is joint work with Rebecca Bellovin and Sean Howe.

15:30-16:00 coffee break

16:00-17:00 Domenico Valloni

Title: p-torsion Brauer group in positive characteristic

Abstract: In this talk, I’ll explain how to build p-torsion elements in the Brauer group using differential forms in characteristic p. We’ll look at some concrete examples to see how the construction works in practice. Then I’ll show how these classes can be used to produce Brauer–Manin obstruction. As an application, we’ll use this to determine the Brauer–Manin set for varieties with ‘many differential forms’.

17:00-18:30 Happy hour