Fibred bundles and group schemes

In this mini-course, we will start, after the first definitions, by introducing the cohomological dimension of a pro-finite group. We shall then concentrate on obtaining explicit bounds in the special case of the étale fundamental group of a curve in positive characteristic. (The first course is entirely based on classical material and should be accessible to advanced graduate students.) In the second lecture, we will investigate other instances of groups of cohomological dimension at most two, mostly through the lens of Galois cohomology. We will review a selection of recent related results and open questions. Material will be taken from my survey paper https://arxiv.org/pdf/2508.09739. In the last lecture, we shall introduce classical (albeit not widespread) techniques in non-abelian (fppf) cohomology, in preparation for G. Bassan's talk.

This three-hour course explores how singularities of algebraic curves give rise to certain group schemes associated to principal bundles. Through the process of pinching a “modulus” (see [1,2,3]), the affine part of a Picard variety of a singular curve can be neatly described: Reduced moduli give rise to toric factors, whereas higher multiplicities produce unipotent groups; in positive characteristic the latter are naturally expressed in terms of truncated Witt groups. Several explicit examples will illustrate how the local geometry of a singularity determines the corresponding affine algebraic group.

This leads naturally to Witt vectors, which will be analysed through the Artin–Hasse exponential [2,4].

By moving to consider principal bundles with non-abelian structure groups (in positive characteristic), we are led to the much less familiar constructions of Ditters and Newman in [5,6]. Their work introduces certain non-commutative analogues of Witt-type Hopf algebras through “divided-power curves” and universal constructions for irreducible cocommutative Hopf algebras. Their results have received comparatively little attention in the subsequent literature, but arise naturally in the context explained before (see [7]) and form a crossroad for the study of Hopf algebras, group schemes and formal groups.

Let X be a smooth projective curve over an algebraically closed field of positive characteristic. The Frobenius pull-back of vector bundles on X induces a rational map between moduli spaces of stable bundles, called the generalized Verschiebung. Understanding the geometry of this map, and in particular computing its generic degree, is a natural problem in the study of vector bundles in positive characteristic. In this mini-course, we will approach this problem through the theory of dormant opers. Roughly speaking, an oper is a flat bundle equipped with a filtration satisfying certain transversality conditions, and in positive characteristic it is called dormant when its p-curvature vanishes. After reviewing dormant opers of higher level and their moduli spaces, we will discuss a factorization property under degeneration of curves, which can be formulated in terms of a two-dimensional topological quantum field theory. This leads to a combinatorial description of dormant PGL(2)-opers in terms of edge numberings on trivalent graphs, providing an effective method for counting them. We will then explain how this enumerative theory is related to the generalized Verschiebung for rank-two stable bundles. In particular, by comparing deformation spaces of maximally Frobenius-destabilized bundles and dormant opers, we can identify the generic degree of the generalized Verschiebung with an appropriate number of dormant opers. Combined with the combinatorial description above, this correspondence yields an explicit procedure for computing the generic degree of the generalized Verschiebung by counting suitable edge numberings. Ultimately, this approach highlights the interplay between the geometry of moduli spaces in positive characteristic and the combinatorics arising from degenerations of algebraic curves.

The problem of understanding extensions of torsors defined over open sets on some larger ambient space has been studied for several years under several guises. In this talk I will discuss a weak notion of extensions of torsors called étale extensions. I will start with the basic definitions and with a local version of the étale extension problem for DVRs. Afterwards, using the local case, I will discuss étale extensions of unipotent torsors over curves, and then discuss their relation to some variants of the fundamental group scheme of a curve.

One of the equivalent conditions for the etate K(Pi,1) property for algebraic varieties is the equality between the etale cohomology with coefficients in etale locally constant torsion sheaves and the cohomology of the etale fundamental group with coefficients in the corresponding representations. We study the similar property for the de Rham cohomology in characteristic zero and the infinitesimal (i.e., stratified) cohomology in positive characteristic. We mainly focus on the case of projective curves.

In 1966, Hartshorne asked whether tensor products and extensions of ample vector bundles remain ample without characteristic or properness assumptions. He proved the tensor-product statement in characteristic zero, and Barton later established it in positive characteristic for proper schemes. I will discuss answers to both questions: tensor products preserve ampleness in arbitrary characteristic (possibly mixed), whereas extensions can fail to be ample on smooth quasi-projective surfaces. The proofs are rather complicated mixture of invariant theory and relative geometric invariant theory. A simple example of bad extension is based on Ejiri-Fujino-Iwai example of a similar sort.

We consider an arbitrary base field k; our object of study are projective algebraic varieties over k, that are homogeneous under a smooth algebraic group. Over the complex numbers, such varieties are direct product of an abelian variety and a generalized flag variety G/P, where G is semi-simple and P is a parabolic subgroup (Borel–Remmert). In positive characteristics, the same decomposition holds (Sancho de Salas); however, the stabilizer P may be non-smooth. This talk will present the structure of non-smooth parabolics over an algebraically closed k. Then we will discuss the generalization of such results to an arbitrary field, which is a work in progress with Michel Brion and Srimathy Srinivasan.

Abstract: When algebraic varieties degenerate, coverings and torsors defined on the smooth locus need not extend across the boundary: ramification and singularities create natural obstructions. Logarithmic geometry provides a framework in which mild ramification and certain singularities behave better, allowing the extension problem to be reformulated in this setting. In this talk, I will explain why extension of torsors can fail in the classical setting and how logarithmic geometry provides a remedy. I will present a logarithmic analogue of the Zariski-Nagata purity theorem which shows that fppf torsors defined away from a divisorial boundary extend uniquely to logarithmic torsors across that boundary, even when the underlying space is singular. I will then explain how the logarithmic extended torsors can be interpreted as ordinary torsors on suitable stacks canonically associated to the base. Finally, I will report on joint work with T. Poiret concerning torsors on degenerating families of nodal curves.

How subtle can the difference between two Galois conjugate complex varieties be? Can they have isomorphic homotopy groups and rational cohomology rings, but nonisomorphic integral cohomology rings? In joint work with Junliang Shen, we show that the answer is yes and the example is given by certain character varieties for curves, answering a 2005 question of Hausel. The proof uses non abelian Hodge theory and develops a systematic method for finding the lowest-degree relation among tautological classes in the cohomology of Higgs moduli spaces, a problem that had previously remained mysterious beyond rank two.