Program

Tuesday, December 11, PK-5115

  • 9:30-10:30 == Campana
  • 10:30-11:00 == Coffee Break
  • 11:00-12:00 == Campana
  • 12:00-14:30 == Lunch
  • 14:30-15:30 == Turchet
  • 15:30-16:00 == Coffee Break
  • 16:00-17:00 == Turchet

Wednesday, December 12, PK-5675

  • 9:30-10:30 == Campana
  • 10:30-11:00 == Coffee Break
  • 11:00-12:00 == Campana

Thursday, December 13, PK-5675

  • 9:30-10:30 == Ascher
  • 10:30-11:00 == Coffee Break
  • 11:00-12:00 == Ascher
  • 12:00-14:30 == Lunch
  • 14:30-15:30 == Campana
  • 15:30-16:00 == Coffee Break
  • 16:00-17:00 == Darondeau

Courses

Arithmetic aspects of orbifold pairs and ‘special’ manifolds.

Speaker: Frédéric Campana

Abstract: One of the major goals of arithmetic geometry is to determine geometric conditions under which a projective manifold defined over a number field is ‘potentially dense’ in the sense that its rational points become Zariski dense after a suitable finite extension of the base field.

The problem is famously solved for curves by knowing the value of the genus g, or equivalently, the sign of the canonical bundle. In particular, for genus 2 the number of rational points defined over any number field is always finite by Faltings’s proof of the Mordell conjecture.

We shall present a conjectural solution to this problem in any dimension by introducing the class of so-called ‘special’ manifolds which are higher-dimensional analogues of rational and elliptic curves (i.e., curves of genus 0 or 1), defined by a certain non-positivity of the cotangent bundle. In short, these special manifolds are conjectured to coincide exactly with the potentially dense manifolds. ‘Specialness’ is alternatively defined by the absence of fibrations onto so-called ‘orbifold pairs’ (X,D) of general type, where D is a divisor of which each component is equipped with a certain positive multiplicity encoding the multiplicities of the fibres.

We will introduce the notion of ‘integral points’ of such orbifold pairs. This allows the extension of the conjectures of Lang and Vojta to this new context.

For curves, this gives an orbifold version of the Mordell conjecture, which is still open, but which follows from the abc conjecture. A weak form has been obtained by Darmon-Granville in 1995, using Falting’s theorem. As a consequence of a well-known conjecture within Vojta’s arithmetic version of Nevanlinna Theory, one can show the implication: X potentially dense implies X not special.

Special manifolds are easy to describe in dimension 2. The situation changes radically in dimension 3, where a construction by Bogomolov-Tschinkel provides non-special examples for which the conjecture above, supported conditionally by Vojta’s theory, conflicts with another conjecture attributed to Abramovich and Colliot-Thélène.

Arithmetic and Geometry of log pairs

Speakers: Ascher-Turchet

Abstract: We present an introduction to the problem of determining arithmetic and geometric hyperbolicity properties of pairs. Log pairs, or pairs, are objects of the form (X,D) where X is a projective variety and D is a reduced divisor. They play a central role both in arithmetic, since they are the right objects to study integral points, and in geometry, especially in the minimal model program and in the study of moduli spaces of higher dimensional varieties. In this mini-course we will introduce pairs and their properties. After reviewing the case of curves, we will present the (few) known results for surfaces. We will then focus on the case of surfaces with positive logarithmic cotangent bundle, describe their geometry and their moduli, and discuss recent proofs of algebraic and arithmetic hyperbolicity.

Orbifold hyperbolicity

Speaker: Darondeau

Abstract: I will present a work with F. Campana and E. Rousseau on the hyperbolicity of geometric orbifolds. I will first recall the notion of entire curve in the category of geometric orbifolds introduced by Campana. I will briefly justify the necessity to work in this setting, which generalizes and extends the classical setting (compact setting and logarithmic setting). We will then see that the natural theory of orbifold jet differentials (that we introduce for higher orders) allows us to obtain new results, but is rather surprising in comparison with the classical setting.