Christophe Soulé

Analytic torsion and holomorphic determinant bundles. II. Direct images and Bott-Chern forms

On etudie les proprietes principales des fibrations de Kahler: On introduit la superconnexion associee de Levi-Civita pour construire des formes torsion analytiques pour des images directes holomorphes

Arithmetic intersection theory

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Graphic Requirements for Multistationarity

We discuss properties which must be satisfied by a genetic network in order for it to allow differentiation. These conditions are expressed as follows in mathematical terms. Let F be a differentiable mapping from a finite dimensional real vector space to itself. The signs of the entries of the Jacobian matrix of F at a given point a define an interaction graph, i.e. a finite oriented finite graph G(a) where each edge is equipped with a sign. René Thomas conjectured 20 years ago that if F has at least two nondegenerate zeroes, there exists a such that G(a) contains a positive circuit. Different authors proved this in special cases, and we give here a general proof of the conjecture. In particular, in this way we get a necessary condition for genetic networks to lead to multistationarity, and therefore to differentiation. We use for our proof the mathematical literature on global univalence, and we show how to derive from it several variants of Thomas rule, some of which had been anticipated by Kaufman and Thomas.

Analytic torsion and holomorphic determinant bundles. I. Bott-Chern forms and analytic torsion

We attach secondary invariants to any acyclic complex of holomorphic Hermitian vector bundles on a complex manifold. These were first introduced by Bott and Chern [Bot C]. Our new definition uses Quillens superconnections. We also give an axiomatic characterization of these classes. These results will be used in [BGS2] and [BGS3] to study the determinant of the cohomology of a holomorphic vector bundle.

Analytic torsion and holomorphic determinant bundles. III. Quillen metrics on holomorphic determinants

In this paper, we derive the main properties of Kähler fibrations. We introduce the associated Levi-Civita superconnection to construct analytic torsion forms for holomorphic direct images. These forms generalize in any degree the analytic torsion of Ray and Singer. In the case of acyclic complexes of holomorphic Hermitian vector bundles, such forms are calculated by means of Bott-Chern classes.

Complex Immersions and Arakelov Geometry

In this paper we establish an arithmetic Riemann-Roch-Grothendieck Theorem for immersions. Our final formula involves the Bott-Chern currents attached to certain holomorphic complexes of Hermitian vector bundles, which were previously introduced by the authors. The functorial properties of such currents are studied. Explicit formulas are given for Koszul complexes.

On the 3-torsion in K4(Z)

Abstract Let SL 4 ( Z ) be the group of four by four integral matrices with determinant one. This group acts upon the top homology of the spherical Tits building of SL4 over Q , i.e. the Steinberg module St4 (see below, 1.2). The goal of this note is to prove the following: Theorem 1. The first homology group H 1 (SL 4 ( Z ), St 4 ) is a finite group of order a power of 2. This result was proved 18 years ago (Soule, These, University of Paris VII, 1979). At the time, I deduced from it that K 4 ( Z ) is the direct sum of a finite 2-group and 0 or Z /3 . Rognes uses Theorem 1 in his proof that K 4 ( Z ) vanishes (J. Rognes, K 4 ( Z ) is the trivial group, Preprint, 1998).

Arithmetic Chow Groups and Differential Characters

This paper sketches the relationship between the arithmetic Chow groups introduced by the authors, and the theory of differential characters due to Cheeger and Simons. Applications given are computing the holonomy of the Quillen connection and studying the Abel-Jacobi homomorphism.

Upper Bounds for Regularized Determinants

Abstract:We conjecture that the zeta-regularized determinant of the Laplace operator with coefficients in a holomorphic vector bundle on a compact Kähler manifold remains bounded when the metric on the bundle varies. This conjecture is shown to be true for certain classes of line bundles on Riemann surfaces.

Secant varieties and successive minima

Given an arithmetic surface and a positive hermitian line bundle over it, we bound the successive minima of the lattice of global sections of this line bundle. Our method combines a result of C.Voisin on secant varieties of projective curves with previous work by the author on the arithmetic analog of the Kodaira vanishing theorem. The paper also includes a result of A.Granville on the divisibility properties of binomial coefficients in a given line of Pascals triangle.