Modul: MAT770 Oberseminar: Algebraische Geometrie

## A differential approach to Ax-Schanuel theorems

Talk by Prof. Dr. Guy Casale

Speaker invited by: Prof. Dr. Rémi Abgrall

**Date:** 12.12.23 **Time:** 13.15 - 14.45 **Room:** Y27H25

Ax-Schanuel theorems are statements about the Zariski closure of a formal (non-constant) curve lying on a leaf of a foliation of a complex algebraic variety. Here are two instances of this family of theorems:

Theorem (Ax): For $t in (C[[s]]-C)^n$, if $trdeg C(t_1,..., t_n, exp(t_1),..., exp(t_n))/C < 1+n$ then a $Z$-linear combination of $t_i$'s is constant.

Theorem (Pila-Tsimerman): For $t in (C[[s]]-C)^n$, if $trdeg C(t_1,...,t_n, j(t_1),..., j(t_n), ..., j''(t_n))/C < 1+3n$ then there exist $k\leq l-1$ and $h$ in $PGL_2^+(Q)$ such that $t_k = h(t_l)$.

The natural extension of these theorems to developing maps of a rational $(G, G/H)$-structure on a algebraic variety splits in two different parts: 1/ prove that $j(t_k)$ is algebraic over $C(j(t_l))$; 2/ prove the existence of $h$. I will explain how the first part can be obtained from a general result on principal connexions using elementary differential Galois theory and how the second part is obtained in the special case of product of projective structures on curves using model theory of differentially closed fields.