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 June 8, the next meeting of the Belgian-Dutch Algebraic Geometry seminar will be held in Leuven. The lectures are given in the Landbouwinstituut Hoofdgebouw, room 00.215 - aula Jozef Heuts

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

13:45-14:45 Arend Bayer (Edinburgh) Families of Hyperkaehler varieties via families of stability conditions
Abstract: Bridgeland stability conditions on derived categories of algebraic varieties and their wall-crossings have recently been used extensively to study the geometry of moduli spaces of stable sheaves. In work in progress with Macri, Lahoz, Nuer, Perry and Stellari, we are extending this toolkit to the "relative" setting, i.e. for a family of varieties. Our construction comes with relative moduli spaces of stable objects; this gives additional ways of constructing new families of varieties from a given family, thereby potentially relating different moduli spaces of varieties. Our main application is for families of cubic fourfolds; in particular, this produces many new examples of algebraically constructed families of Hyperkaehler varieties over a base of maximal dimension 20.
15:00-16:00 Christian Liedtke (Technische Universität München) A Néron-Ogg-Shafarevich criterion for K3 Surfaces
Abstract: Let R be complete local ring with field of fractions K and residue field k, let X be a K3 surface over K, and assume that X has potential semi-stable reduction (which is automatic if char(k)=0). If char(K)=0, then we show that the following are equivalent: (1) the l-adic Galois representation on H2(X,Ql) is unramified for one l different from p; (2) the l-adic Galois representation on H2(X,Ql) is unramified for all l different from p; (3) the p-adic Galois representation on H2(X,Qp) is crystalline; (4) the surface has good reduction after an unramified extension of K.
For example, if R is a power series over the complex numbers, then this says that a family of K3 surfaces over a pointed disk can be filled in smoothly over the origin (that is, has good reduction) if and only the monodromy representation on H2 is trivial. However, in the arithmetic situation, where the residue field k might not be algebraically closed, then an unramified base change might be needed. We show by example that sometimes, a non-trivial base-change is necessary. In any case, we show that if (1) to (3) hold, then there always exists a model over R whose special fiber X0 has at worst canonical singularities. Then, good reduction of X is equivalent to having an isomorphism between H2 of X and the minimal resolution of singularities of X0, such that this isomorphism is compatible with the natural Galois-actions (or F-isocrystal structures). In my talk, I will introduce all the above notions, which will not give me much time to explain proofs. Part of this is joint with Matsumoto, part of this is joint with Chiarellotto and Lazda.
16:30-17:30 Lionel Darondeau (KUL) Jet differentials and hyperbolicity
Abstract: Hyperbolicity is the study of the geometry of holomorphic entire curves f: C→ X with values in a given complex manifold. In this talk, we will give some definitions and provide motivating historical examples. Then, we will describe the formalism of jets, that can be viewed as a coordinate free description of the differential equations that entire curves may satisfy, and explain a successful general strategy to study hyperbolicity due to Bloch, Demailly, Siu. We will end the talk with a quick review of some recent results on hyperbolicty and related topics.

Previous meetings of the seminar (in reverse chronological order)

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
José Burgos Gil (ICMAT, Madrid) Grothendieck's period conjecture and zeta values

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