Differential forms in algebraic geometry

9-10 Shunsuke Takagi
F‑singularities are singularities in positive characteristic defined using the Frobenius morphism. F‑singularities conjecturally correspond via reduction modulo p to singularities appearing in birational geometry in characteristic zero. In this talk, I survey recent developments on this conjectural correspondence. I also discuss a relationship between Frobenius splitting and log Calabi-Yau varieties, which can be viewed as a global version of the above correspondence.

11-12 Arne Smeets
Report on joint work with Stefan Kebekus. I will explain how to formulate a version of Mordell's conjecture for Campana's "orbifold pairs". I will then show how to prove analogues, for orbifold curves over function fields, for the classical geometric height inequalities obtained by Vojta (in characteristic 0) and Kim (in positive characteristic). As an application, I will present the first (unconditional) examples of simply connected varieties over global fields for which the Brauer-Manin obstruction does not explain the failure of the Hasse principle.

14-15 Joseph Steenbrink
Abstract TBA

16-17 Yohan Brunebarbe
We show that all subvarieties of a quotient of a bounded symmetric domain by a sufficiently small arithmetic discrete group of automorphisms are of general type. In particular, we can show with our approach that for any g ≥ 1 and any n > 12.g, every subvariety of the moduli space of principally polarized abelian varieties of dimension g with a level-n structure is of general type.

9-10 Hélène Esnaualt
This talk is about joint work with T. Abe.

11-12 Zsolt Patakfalvi
I will explain some of the results and techniques developed in the past few years jointly with Christopher Hacon to study the behavior of the Albanese morphism of varieties over (perfect) fields of positive characteristic. The main tool is a generic vanishing type theorem for Cartier modules developed also during this project.

14-15 Takao Yamazaki
Using smooth proper curves over Q with two disjoint effective divisors and a notion of de Rham cohomology for such "curves with modulus", we construct an Abelian category and show that it is naturally equivalent to the category of Laumon 1-isomotives. This result extends and relies on the theorem of J. Ayoub and L. Barbieri-Viale that describes Deligne's category of 1-isomotives in terms of Nori's category of motives. This is joint work with F. Ivorra.

16-17 Kevin Tucker
The F‑signature is a numerical invariant of singularities which measures the asymptotic number of splittings of iterates of Frobenius. The positivity of the F‑signature characterizes F‑regular singularities, which are closely related to KLT singularities in characteristic zero. After giving an overview, I will discuss new transformation rules for F‑signature under finite maps. These transformation rules allow us to show finiteness of the etale local fundamental group for F‑regular singularities, analogous to results of Xu and Greb-Kebekus-Peternell for KLT singularities in characteristic zero.

9-10 Adrian Langer
I will survey some results concerning various conjectural characterizations of motivic representations of the topological fundamental group of a smooth complex quasi-projective variety. In particular, I plan to prove that rank 3 integral rigid representations come from some families of smooth projective varieties. The proof uses characterization of varieties whose differential forms contain some interesting rank 1 subsheaves. This is a joint work with C. Simpson.

11-12 Joseph Ayoub
The foliated topology is a direct analog of the etale topology in the category of schematic foliations. In the same way that etale topology is related to Galois theory, foliated topology is related to differential Galois theory. It also allows one to obtain higher differential Galois groups associated to differential fields. This is a new phenomenon as abstract fields have no higher Galois groups.

9-10 Jean‑Louis Colliot‑Thélène
Abstract TBA

11-12 Sándor Kovács
This is a report on joint work with Christopher Hacon. We give a new proof of Alexeev's boundedness result for stable surfaces which is independent of the base field. We also highlight some important consequences of this result.

14-15 Daniel Greb
By a famous theorem of Beauville, Bogomolov, and Fujiki, an "indecomposable" projective manifold with trivial canonical bundle is either an Abelian variety, a (simply-connected) Calabi-Yau manifold, or an irreducible holomorphic-symplectic manifold (also called hyperkähler manifold). These three classes can be distinguished by looking at their algebra of holomorphic differential forms. In the Minimal Model Program, we are forced to work with singular varieties, to which the differential-geometric techniques used in establishing the above results do not apply directly. I will discuss the question of how to generalise the notion "indecomposable" to the singular setup, introduce the three "obvious" classes of indecomposable singular varieties with trivial canonical class corresponding to the smooth ones listed above, and present the techniques and ideas used in the proof of the fact that also in the singular case the three "obvious" classes exhaust all possibilities of indecomposable singular varieties with trivial canonical class. This is work in progress with Henri Guenancia and Stefan Kebekus.

16-17 Kay Rülling
I will recall the definition of motivic cohomology of a modulus pair as introduced by Binda-Saito and the Kato-Saito relative Milnor K-theory. We will see that these are closely related. If time permits, I will explain the relation with geometric global class field theory and the connection with differentials. This is joint work with Shuji Saito.

9-10 Florian Ivorra
In this talk I will explain how one may use the tannakian approach of motives due to Madhav Nori to develop a theory of perverse motivic sheaves over any separated scheme of finite type over a field of characteristic zero embedded into the field of complex numbers. I will also relate the derived category of this Abelian category of perverse motivic sheaves to the triangulated category of étale motives of Morel-Voevodsky. This provides in particular a Hodge realization functor to the derived category of mixed Hodge modules introduced by Morihiko Saito.

11-12 Veronika Ertl
This is a report on work in progress. In characteristic 0 Huber and Jörder gave a very conceptual approach to differential forms using the h-topology. In positive characteristic, it is natural to look at the de Rham-Witt complex as a replacement for the usual de Rham complex. We study Witt differentials using a similar approach as Huber and Jörder while avoiding resolution of singularities. In particular we are interested in descent statements, and for this use techniques developed in work by Huber-Kebekus-Kelly.

14-15 Karl Schwede
I will talk about joint work with Ma and Shimomoto where we study the dualizing complexes (resp. local cohomology) of Du Bois singularities and their thickenings. As consequences, we obtain results about Du Bois singularities in families and show that F‑injective singularities deform for p large.