Abstracts, exercises

Lecture series:

Guy Casale (Rennes) and Remi Jaoui (Freiburg)

Title: A differential approach to Ax-Schaunel Theorems.

Abstract: Ax-Schanuel Theorems describe the algebraic relations between different evaluations of a given function f along different formal power series. In these lectures we are interested in the case f is a "uniformizer" of a rational (G,X)-structure on an algebraic manifold.

Lecture 1: Differential Galois theory and algebraic dependence

We will present the basic definitions and theorems from differential Galois theory and the so-called Goursat-Kolchin Lemma.

Lecture 2: Special subvarieties of a integrable principal connection.

Special subvarieties are subvarieties such that the restriction of the connection has a smaller Galois group. We will present a theorem about subvariety V of the principal bundle and horizontal leaf L of the connection such that codim (L cap V) < codim L + codim V. Under suitable hypothesis the projection of the intersection in the base manifold is included in a special subvariety.

Lectures 3 and 4: The case of a simple rational projective structure on a curve.

A projective structure on a curve is a second order differential equation and a uniformizer is given by the inverse of the quotient of two solutions (it may not be a uniforme surjective map).

Using the previous theorem and Goursat-Kolchin, we will prove that the existence of enough algebraic relations between evaluations of the unifomizer along formal power series implies existence of algebraic relations between two of them.

Using tools from model theory of differentially closed fields, we will give a precise desctiption of these relations.

This extends previous work of J. Pila and J. Tsimermann of the j function.

These lectures are based on https://arxiv.org/abs/2102.03384 with D. Blazquez-Sanz, J. Freitag and R. Nagloo.

Exercises

Paolo Cascini (Imperial) and Calum Spicer (King's College)

Title: Minimal Model Program for foliations.

Abstract: The lecture series will be based on https://arxiv.org/abs/1808.02711, https://arxiv.org/abs/2012.11433, and https://arxiv.org/abs/2111.00423.

Exercises

Research talks:

Carolina Araujo (IMPA)

Title: Foliations with positive tangent sheaves and characterizations of projective space bundles.

Abstract: The existence of sufficiently positive subsheaves of the tangent bundle of a complex projective manifold X imposes strong restrictions on X. In particular, several special varieties can be characterized by positivity properties of their tangent bundle. In this talk, we will discuss various notions of positivity for foliations on complex projective manifolds, and the effect of positivity on the algebraicity of their leaves. We then focus on foliations having big slopes with respect to movable curve classes, obtaining characterizations of projective space bundles, and a lower bound for the algebraic rank of a foliation in terms of invariants measuring positivity. This is a joint work with Stéphane Druel.

André Belotto da Silva (Paris Cité)

Title: Splittable foliations and the minimal rank Sard Conjecture.

Abstract: TBA.

Jarosław Buczyński (IMPAN)

Title: Characteristic foliations on contact manifolds.

Abstract: A holomorphic symplectic manifold is a complex manifold endowed with a closed nowhere degenerate 2-form (called a symplectic form on Y). As a standard fact, such manifold Y necessarily has even dimension. Consider any irreducible divisor D in Y. Such divisor naturally comes with a one-dimensional foliation (called a characteristic foliation of D) determined generically by the perpendicular (with respect to the symplectic form) spaces to the tangent spaces TD. In the talk I am interested in an odd dimensional analogue of symplectic manifolds, namely in contact manifolds. A complex manifold X is a contact manifold if the tangent bundle TX is equipped with a codimension 1 distribution which is as far from being a foliation as possible (formally, its O'Neil tensor is nowhere degenerate). In an analogous way as in the symplectic case, we can define the characteristic foliation for any divisor in X. A major open problem is to classify all contact Fano manifolds (conjecturally, they are all homogeneous spaces G/P, for a specific parabolic subgroup P of a simple Lie group G). Some partial results in this direction have been obtained by analysing specific divisors on contact Fano manifolds and "guessing" their integral curves.

Jorge Pereira (IMPA)

Title: Codimension one foliations in positive characteristic.

Abstract: I will discuss the geometry of codimension one foliations in positive characteristic and explain how it can be used to provide new information on the space of codimension one foliations on projective spaces. The talk will be based on joint work with Wodson Mendson (https://arxiv.org/abs/2207.08957).

Frédéric Touzet (Rennes)

Title: Numerically nonspecial varieties (joint work with Erwan Rousseau and Jorge Pereira).

Abstract: Campana introduced the class of special varieties as the varieties admitting no Bogomolov sheaves i.e. rank one coherent subsheaves of maximal Kodaira dimension in some exterior power of the cotangent bundle. Those are precisely varieties which do not admit any surjective map onto a general type orbifold. Campana raised the question if one can replace the Kodaira dimension by the numerical dimension in this characterization. We answer partially this question showing that a projective manifold admitting a rank one coherent subsheaf of the cotangent bundle with numerical dimension 1 is not special. We also establish the analytic characterization with the non-existence of Zariski dense entire curve and the arithmetic version with non-potential density in the function field setting.