# 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

Abstract: 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 H1 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

Abstract: We state the main theorem and we explain how to get a lower bound for the first minimum of H1. 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

Abstract: 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 H1. In our conclusion, we mention other inequalities satisfied by successive minima on curves and surfaces of general type.
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