# 2009 Bowen Lectures

*The Twenty-Ninth Annual Bowen Lectures will be delivered by Christophe Soulé on November 17, 18 and 19.*

Department of Mathematics, University of California, Berkeley, presents

The 2009 Bowen Lectures

Christophe Soulé

Institut des Hautes Études Scientifiques (IHES)

**Arithmetic surfaces and successive minima**

*Abstract:**Arithmetic surfaces*are models over the integers of smooth projective curves over the rationals. They play a central role in Vojta’s proof of the Mordell conjecture (first solved by Faltings). Today, a main open problem is to get an effective proof of that theorem. According to Parshin and Moret-Bailly, this could be done if one knew a good upper bound for the Arakelov self-intersection of the dualizing sheaf. In these lectures I will prove a series of inequalities which bound this number by the successive minima of a euclidean lattice.

**Lecture 1**: Minkowski’s theorem and arithmetic surfaces

November 17: 4:00-5:00 pm , Room 50, Birge Hall

**To a euclidean lattice, i.e. a free abelian group of rank N equipped with a scalar product on its real span, Minkowski associated an increasing sequence of N real numbers called the successive minima of the euclidean lattice. Their sum is essentially the logarithm of the covolume of the lattice. Given an arithmetic surface X and a line bundle L over X equipped with an hermitian metric on the corresponding holomorphic line bundle, our goal in these lectures will be to estimate from below the successive minima of the group H**

*Abstract:*^{1}of extensions of L by the trivial line bundle. These bounds will be expressed as Arakelov intersection numbers over X.

**Lecture 2**: A vanishing theorem in Arakelov geometry

November 18: 4:00-5:00 pm , Sibley Auditorium

**We state the main theorem and we explain how to get a lower bound for the first minimum of H**

*Abstract:*^{1}. The proof follows closely Mumford’s proof of the Kodaira vanishing theorem for algebraic surfaces.

**Lecture 3**: Secant varieties and successive minima

November 19, 4:00-5:00 pm , Location: Room F295, Haas

**Given a projective curve C over the complex numbers, we consider a sequence of varieties called the secant varieties of C . A theorem of Voisin bounds the dimension of the linear subspaces of these secant varieties. Using this theorem, we extend the arguments of lecture 2 to get lower bounds for higher successive minima of H**

*Abstract:*^{1}. In our conclusion, we mention other inequalities satisfied by successive minima on curves and surfaces of general type.