Talks

The original Riemann–Hilbert correspondence is a fundamental theorem asserting that, for a smooth complex variety X, there is an equivalence between a suitable class of D-modules (namely, modules equipped with an action of differential operators) and complex constructible sheaves in the analytic topology. It has found numerous applications to this day, ranging from the study of differential equations, singular varieties and their cohomology, fibrations, representation theory, to the development of the theory of Hodge modules.

Our objective in this lecture series is to explore avatars of this equivalence in positive characteristic, thereby relating modules with Frobenius actions/Cartier operators and F_p-sheaves in the appropriate topology. We will start with presenting the pioneering work of Emerton–Kisin on the subject, then move to the theory developed by Böckle–Pink, which was more recently revisited by Bhatt–Lurie. Finally, we will talk about new generalizations, which are joint work with Kay Rülling, Jakub Witaszek and Bogdan Zavyalov. As we explore the different theories, we will discuss their respective applications.

The classical de Rham theorem which compares Betti cohomology and de Rham cohomology for a smooth projective variety over the complex numbers has inspired several comparison conjectures in the p-adic world [Fo94], which over the course of many years have become theorems due to the work of many people like Fontaine–Messing, Hyodo, Kato, Faltings, Tsuji, Nizioł, Beilinson, Bhatt, Scholze. In fact, a whole theory has developed out of this, called p-adic Hodge theory, with many recent developments in the realm of rigid analytic geometry.

One important building block of p-adic Hodge theory is Hyodo-Kato theory [HK94]. Taking a historical perspective, I will explain the concept of Hyodo-Kato theory, and explain what role it plays in the above mentioned comparison theorems.

Then I will explain the classical construction due to Hyodo and Kato and if time permits a rigid analytic approach to Hyodo-Kato theory (which is joint work with Kazukia Yamada), which gives a very explicit construction, and explain how it related with other (more abstract) constructions.

Etale cohomology groups in positive characteristic~p with p-torsion coefficients tend to be quite ill behaved when it comes to finiteness, homotopy invariance, and cohomological purtiy. This minicourse presents a modification of the étale topology, called tame topology, that overcomes these issues.

We start by studying tame covering spaces of schemes~X of finite type over a base space~S. There are several approaches in the literature starting from tameness along a divisor and including curve tameness and divisor tameness. We obtain the strongest and most flexible concept by looking at the adic space Spa(X,S) and studying its finite étale coverings that are tame at every point. It allows for generalizations to more general adic spaces such a rigid spaces and gives a well behaved tame fundamental group also in this setting.

The viewpoint of adic spaces allows not only the construction of a tame fundamental group via tame covering spaces but also the definition of a tame site consisting of étale morphisms that are tame at every point. We show that the resulting cohomology groups with p-torsion coefficients in characteristic~p satisfy the properties expected from a "motivic" cohomology theory in contrast to the respective étale cohomology groups.

We give an overview on the basic definitions and structural results on reciprocity sheaves on schemes of positive characteristics, with a focus on the motivic nature of their ramification filtrations and their "tame part".

An important problem in birational geometry is trying to relate in a meaningful way the canonical bundles of the source and the base of a fibration. The canonical bundle formula addresses this problem: it describes the relation between the canonical bundles in terms of the singularities of the fibres and a divisor measuring their variation in moduli.

Contrary to complex fibrations, in positive characteristic even if the source and the base are smooth, the fibres can be highly singular. In this talk we will see in some threefold examples how we can have a weak canonical bundle formula for these singular fibrations.

This is based on work in progress joint with J. Kountouridis

We define privileged local systems, a generalisation in any dimension of Katz’ notion of physically rigid local systems on an open of the projective line and prove the p-curvature conjecture for them. 

Let X be a smooth and proper variety over an algebraically closed field k of characteristic p. It is well-known that the \ell-primary torsion of the Brauer group of X contains a maximal divisible subgroup, such that the quotient is a finite torsion \ell-group. On the other hand, the quotient of the p-primary torsion by its maximal divisible subgroup behaves much more wildly: it is of the form U(k), where U is an extension of a finite group by a unipotent k-group scheme, and thus it is often of infinite type. This reflects the peculiar behaviour of the fppf cohomology groups of Z_p(1). In this talk we will explain how to study these groups via the crystalline cohomology of X and the Nygaard filtration, and we will give some results on the shape of U when X is an abelian variety. Time permitting, we will briefly discuss how these groups vary in families. Joint work with Alexei Skorobogatov and Yuan Yang.

Local stability is a local singularity condition for families of varieties over curves, going back to the work of Kollár and Shepherd–Barron. It is formulated in the language of the Minimal Model Program, and turned out to be extremely useful for the compactification of moduli spaces in characteristic zero. The same definition can be cast in positive characteristic, but in this setting many properties become difficult to prove. In this talk I will discuss the following basic question: is local stability (in positive characteristic) preserved by base-change? I will explain how one can reduce to base-change by Frobenius morphisms, and which evasive (at least for the time being!) invariants control the situation. This is partly based on joint work with Marta Benozzo.

This talk will first discuss some results on étale fundamental groups of varieties over an algebraically closed field of characteristic p > 0, based on joint work with Hélène Esnault and other coauthors. One result, along with Mark Schusterman, is that the tame fundamental group is finitely presented for such a variety which is the complement of an SNC divisor in a smooth projective variety. A second, along with Jakob Stix, is to give an obstruction for a smooth projective variety to admit a lifting to  characteristic 0, in terms of the structure of its etale fundamental group as a profinite group. We will then touch on some other related work, and on some open questions.

In the 1980s, Mori–Mukai completed the classification of smooth Fano threefolds in characteristic zero, based on earlier work by Iskovskih and Shokurov. Recently, the classification has been extended to the case of positive characteristic. In this talk, we overview some ideas of the proofs and the current status of smooth Fano threefolds in positive characteristic.