This is the homepage of a 2-day conference in Nijmegen on arithmetic geometry. The conference starts on Wednesday September 23rd and ends on Thursday September 24th with the PhD defense of Finn Bartsch. The schedule is below; note that the room changes with every session.
About the rooms: Linnaeus is Heyendaalseweg 137, Huygens is Heyendaalseweg 135, and the Aula is Comeniuslaan 2.
Speakers: Daniel Greb, Frederic Campana, Damian Brotbek, Erwan Rousseau, Claire Voisin
Titles and abstracts below
Wednesday 10h-10:30h: Welcome
Room: Just outside of the room LIN2 in the Linnaeus building
Wednesday 10:30h - 11:30h: Erwan Rousseau
Room: LIN2 (Linnaeus building)
Wednesday 12h - 13h : Daniel Greb
Room: LIN2 (Linnaeus building)
Wednesday 13h-15h: lunch break at the Huygensgebouw. Lunch is taken care of.
Wednesday 15h-15:30h: coffee
Room: Just outside of HG00.308 in Huygensgebouw
Wednesday 15:30h-16:30h: Damian Brotbek
Room: HG00.308 (Huygensgebouw)
Wednesday 17h-18h: Frederic Campana
Room: HG00.308 (Huygensgebouw)
Thursday 10h-1030h: Coffee
Room: Just outside of HG00.307 (Huygensgebouw)
Thursday 1030h-1130h: Claire Voisin
Room: HG00.307 (Huygensgebouw)
11:30h : Walk towards de Aula. Small lunch is served there. This is a 15 minute walk approximately.
12:30h : PhD defense Finn Bartsch
Location: De Aula
Damian Brotbek
Title: Surfaces of general type with extremal cotangent dimension
Abstract: An important result of McQuillan implies that surfaces with big cotangent bundle don’t contain any Zariski dense entire curve. It follows from the Riemann-Roch Theorem that surfaces for which the Chern numbers satisfy c_1^2>c_2 have big cotangent bundle. But in general, a surface can have big cotangent bundle without satisfying the above inequality of Chern numbers, which raises the question of studying the geography of surfaces with big cotangent bundle, namely to study which Chern numbers can be realized by surfaces with big cotangent bundle, and in particular to look for the lowest possible bound on the slope c_1^2/c_2 . Many such examples have been constructed over the years. In this talk, we will explain the situation of Horikawa surfaces, which are the surfaces of lowest possible slope among minimal surfaces of general type, and we will prove that on the one hand, generic Horikawa surfaces admit no symmetric differential forms, but that on the other hand, there exists examples of Horikawa surfaces with big cotangent bundle. This is a joint work with Bruno DeOliveira and Erwan Rousseau.
Frederic Campana
Title: Strictly Simple Fourfolds
Abstract: A compact connected Kahler manifold X of dimension n at least 2 is said (after A. Fujiki) to be "simple" (respectively, "strictly simple") if the only subvarieties through a general (respectively, any) point x in X are x and X. Simpleness is thus the exact opposite of projectivity. X is "Kummer" if it is bimeromorphic to T/G, where T is a torus and G is a finite group. General tori and K3 surfaces are strictly simple. General hyperkahler manifolds and Kummer manifolds are simple. We shall briefly explain how any compact Kahler manifold X is functorially decomposed in terms of simple, Kummer, and projective manifolds by means of five canonical fibrations.
Conjecture: Simple manifolds are either Kummer or bimeromorphically symplectic, that is, even-dimensional with H^(2,0)(X) generated by a form s whose maximal exterior power is nonzero. Odd-dimensional simple manifolds should thus be Kummer.
In particular, strictly simple manifolds should be either etale quotients of a torus or hyperkahler. This is known in dimensions 2 and 3.
Theorem: Strictly simple fourfolds are either etale quotients of a torus or hyperkahler.
We shall sketch the proof.
Daniel Greb
Title: Arakelov inequalities and characterisation of totally geodesic ball quotients
Abstract: I will talk about joint work with Carolina Tamborini and Matteo Costantini, in which we establish an Arakelov inequality for variations of Hodge structures underlying families of principally polarised complex Abelian varieties. In the equality case, this leads to a numerical characterisation of certain totally geodesic ball quotients inside the moduli space of Abelian varieties. Our result extends work of Möller-Viehweg-Zuo by removing the strong positivity conditions on the log-canonical and the log-cotangent bundle imposed in their statement. For families over compact base spaces, our proof involves showing in a first step that the period map associated with a family of Abelian varieties factors through certain operations of the MMP and then generalising the results of Möller, Viehweg, and Zuo to the resulting singular setting. In the quasiprojective case, we develop a new approach and execute it in dimension two: it first descends the family to a very specific partially ample model and then uses symmetric space theory and semistability considerations to detect totally geodesic ball quotients in the moduli space considered with the locally symmetric orbifold Kähler metric.
Erwan Rousseau
Title: Generalized Bloch-Ochiai’s theorem
Abstract : The classical Bloch-Ochiai’s theorem states that a projective manifold with irregularity larger than its dimension has no Zariski dense entire curve.
In joint works with S. Kebekus and F. Touzet, we consider generalized irregularities. I will describe their main properties, relations with specialness (in the sense of Campana) and new applications to hyperbolicity questions.
Claire Voisin
Title: On the Chow ring of very general abelian varieties and a question of Pirola
Abstract: On the moduli space of genus 4 curves, there is a rational section of the Kummer fibration (that is, the Jacobian fibration mod. +- Id) given by the difference of the two trigonal divisors. We prove that any rational section of the Kummer fibration is a multiple of this section, solving in the affirmative a question asked by Pirola. A first step for the proof is a result concerning the Chow ring of a very general abelian variety A of dimension at least 4, or very general Jacobian A in genus at least 4 . We prove that a divisor D on A that satisfies $D2=0$ in $CH2(A)$ must be of torsion.