Organizers: Arne Smeets and Ben Moonen (Nijmegen), Johannes Nicaise (Imperial College/Leuven), Lenny Taelman (Amsterdam), Nero Budur (Leuven) and Bas Edixhoven (Leiden)

June 8, 2018 at the KU Leuven

On September 20 and 21, the next meeting of the Belgian-Dutch Algebraic Geometry seminar will be held at Radboud University Nijmegen. A special 2-day event this time! The lectures are given in the Huygensgebouw (Heijendaalseweg 135), room HG00.071.

Programme (click a title to show/hide the abstract)

Thursday, 20 September
13:30-14:30 Ching-Li Chai (University of Pennsylvania) Sustained p-divisible groups and the foliation on moduli spaces of abelian varieties
Abstract: We will explain joint work with Frans Oort on the notion of 'sustained p-divisible groups' and its application to the foliation structure on moduli spaces of abelian varieties in positive characteristic p. This notion is a scheme-theoretic version of the concept of 'geometrically fiberwise constant p-divisible group' introduced by Frans Oort in 2001 in order to define the central foliation structure on moduli spaces of abelian varieties. Thus we get a scheme-theoretic definition of central leaves in moduli spaces of abelian varieties. This updated definition facilitates the analysis of differential properties of central leaves and reveals their fine local structure. If time permits we will also discuss recent progress on local rigidity for formal completions of central leaves.
15:00-16:00 Jörg Wildeshaus (Université Paris 13) Weights and conservativity
Abstract: After an introduction to weight structures, and a comparison to t-structures, we shall concentrate on conservativity of functors whose source is a triangulated category with a weight structure. Two questions will be of interest to us: (1) Does conservativity of the restriction of the functor to the heart of the weight structure imply conservativity of the functor itself? (2) Is it possible to control the weights of an object from its image under the functor (this principle will be referred to as "weight conservativity")? It turns out that (1) and (2) admit a common answer. It is based on the notion of "minimal weight filtration". Our main application concerns the étale realization of motives of Abelian type.
16:30-17:30 Kestutis Cesnavicius (CNRS / Université Paris-Sud) Purity for the Brauer group
Abstract: A purity conjecture due to Grothendieck and Auslander-Goldman predicts that the Brauer group of a regular scheme does not change after removing a closed subscheme of codimension at least 2. The combination of several works of Gabber settles the conjecture except for some cases that concern p-torsion Brauer classes in mixed characteristic (0,p). We will discuss an approach to the mixed characteristic case via the tilting equivalence for perfectoid rings.
Friday, 21 September
09:30-10:30 Claire Voisin (Collège de France) Gonality and zero-cycles of abelian varieties
Abstract: The gonality of a variety is defined as the minimal gonality of a curve sitting in the variety. We prove that the gonality of a very general abelian variety of dimension g goes to infinity with g. We use for this a generalization of a method of Pirola that we will describe, and rests on the notion of naturally defined subsets of abelian varieties. The method also leads to a number of other applications concerning 0-cycles modulo rational equivalence on very general abelian varieties. For example, for a general abelian variety A of dimension at least 2k – 1, the orbit of k{x} is countable for any point x of A. Furthermore, the set of divisors D such that Dk = 0 in CHk(A) is countable.
11:00-12:00 Victoria Hoskins (Freie Universität Berlin) On the motive of the stack of vector bundles on a curve
Abstract: Following Grothendieck's vision that a motive of an algebraic variety should capture many of its cohomological invariants, Voevodsky introduced a triangulated category of motives which partially realises this idea. After describing some of the properties of this category, I will explain how to define the motive of certain algebraic stacks by generalising a construction of Totaro. I will then focus on defining and studying the motive of the moduli stack of vector bundles on a smooth projective curve and show that this motive can be described in terms of the motive of this curve and its symmetric powers. If there is time, I will give a conjectural formula for this motive, and explain how this follows from a conjecture on the intersection theory of certain Quot schemes. This is joint work with Simon Pepin Lehalleur.
13:00-14:00 Richard Thomas (Imperial College London) Refined Vafa-Witten invariants for projective surfaces
Abstract: I'll describe how to define Vafa-Witten invariants – and refined Vafa-Witten invariants – of projective surfaces. Then I'll explain how to calculate some parts of the theory via degeneracy loci and "Carlsson-Okounkov operators". The talk describes different projects joint with Yuuji Tanaka, Davesh Maulik and Amin Gholampour.
14:30-15:30 Piotr Achinger (IMPAN, Warsaw) Serre-Tate theory for Calabi-Yau varieties
Abstract: Classical Serre–Tate theory describes deformations of ordinary abelian varieties. It implies that every such variety has a canonical lift to characteristic zero and equips the moduli space with a Frobenius lifting and canonical multiplicative coordinates. I will discuss a construct canonical liftings modulo p^2 of varieties with trivial project, joint with Maciej Zdanowicz (EPFL), whose aim is canonical class which are ordinary in the weak sense that the Frobenius acts bijectively on the top cohomology of the structure sheaf. Consequently, we obtain a Frobenius lifting on the moduli space of such varieties. The quite explicit construction uses Frobenius splittings and a relative version of Witt vectors of length two.

Previous meetings of the seminar (in reverse chronological order)

June 8, 2018 at the KU Leuven

Arend Bayer (Edinburgh) Families of Hyperkaehler varieties via families of stability conditions
Christian Liedtke (Technische Universität München) A Néron-Ogg-Shafarevich criterion for K3 Surfaces
José Burgos Gil (ICMAT, Madrid) Grothendieck's period conjecture and zeta values

March 23, 2018 at Leiden University

Videos of the lectures are available via youtube.
Florian Ivorra (FRIAS, Freiburg) Quasi-unipotent motives
Peter Jossen (ETH Zürich) Galois groups and Motivic Fundamental groups
Lionel Darondeau (KU Leuven) Jet differentials and hyperbolicity

December 15, 2017 at Radboud University Nijmegen

Johannes Nicaise (Imperial College and Leuven) Specialization of (stable) rationality in families with mild singularities
Javier Fresán (École Polytechnique) Exponential motives
Michael Groechenig (FU Berlin) p-adic integration for the Hitchin system

December 9, 2016 at the University of Amsterdam

June Huh (Princeton) Enumeration of points, lines, planes, etc.
Dieter Kotschick (München) and Stefan Schreieder (Bonn) The Hodge ring of Kähler manifolds
Bhargav Bhatt (Ann Arbor) Derived splinters in mixed characteristic

April 22, 2016 at KU Leuven

Sho Tanimoto (Copenhagen) Towards a refinement of Manin's conjecture
John Christian Ottem (Oslo) Effective cones of cycles on blow-ups of projective space
Bruno Chiarellotto (Padova) Monodromy action and special fiber for semistable schemes over a DVR

December 11, 2015 at Leiden University

David Holmes (Leiden) Models of degenerating jacobians
Clément Dupont (MPIM Bonn) Motives of bi-arrangements
Misha Verbitsky (ULB, Bruxelles) Transcendental Hodge algebra

June 11 and 12, 2015 in Nijmegen, joint with NoGAGS

Daniel Greb (Duisburg-Essen) Movable curves and semistable sheaves
Klaus Künnemann (Regensburg) A tropical approach to non-archimedean Arakelov geometry
Kevin Buzzard (Imperial College) Pre-adic spaces and adic spaces
Tim Dokchitser (Bristol) Local arithmetic of hyperelliptic curves
David Rydh (KTH Stockholm) The local structure of Artin stacks
Lars Kindler (FU Berlin) Ramification theory for D-modules in positive characteristic
Christian Lehn (Hannover) Rational curves on singular cubic fourfolds
Ulf Kühn (Hamburg) A geometric approach to constructing elements of K2 of curves

November 14, 2014 at the University of Amsterdam

Minhyong Kim (Oxford) Non-abelian reciprocity laws and Diophantine geometry
Mingmin Shen (UvA) Multiplicative Chow-Künneth decompositions
Jakob Stix (Frankfurt) Gorenstein orders and abelian varieties over finite fields

May 16, 2014 in Leuven

Olivier Benoist (Strasbourg) Complete families of smooth space curves
Frank-Olaf Schreyer (Saarbrücken) Matrix factorizations and families of curves of genus 15
Carel Faber (Utrecht) On the cohomology of the moduli spaces of pointed curves of genus three

November 22, 2013 in Nijmegen

Nero Budur (Leuven) Cohomology jump loci
Stefan Schröer (Düsseldorf) Wild quotient surface singularities
René Schoof (Rome) Finite group schemes and abelian varieties with good reduction outside one prime

June 14, 2013 in Leiden

Walter Gubler (Regensburg) Normal toric varieties over valuation rings of rank 1
Gaël Rémond (Bordeaux) Isogenies and polarizations
Jean-Benoît Bost (Orsay) Pluripotential theory, heights, and algebraization on arithmetic threefolds
José Ignacio Burgos Gil (ICMAT Madrid) The singularities of the invariant metric of the sheaf of Jacobi forms on the universal elliptic curve

April 19, 2013 in Leuven

Alejandro Soto (Regensburg) Toric Varieties over a Valuation Ring
Sébastien Boucksom (Jussieu) A uniform version of Izumi's theorem
Ariyan Javanpeykar (Leiden) Bounds for Arakelov invariants - algorithmic and Diophantine applications

December 7, 2012 at the UvA

Andrei Caldararu (Wisconsin) Derived intersections
Eduard Looijenga (Utrecht) Cohomological dimension of moduli spaces of curves
Daniel Huybrechts (Bonn) Cycles on K3 surfaces

June 15, 2012 in Leuven

Ted Chinburg (UPenn) Small generators for S-arithmetic groups
Matthieu Romagny (Jussieu) Models of groups schemes of roots of unity
Mircea Mustaţă (Michigan) Adjoint line bundles in positive characteristic