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Christophe Soule

Christophe Soule
CNRS · IHES

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72
Publications
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4,347
Citations
Citations since 2016
0 Research Items
1008 Citations
2016201720182019202020212022050100150
2016201720182019202020212022050100150
2016201720182019202020212022050100150
2016201720182019202020212022050100150
Introduction
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Publications

Publications (72)
Article
For , 6 and 7, using the classification of perfect quadratic forms, we compute the homology of the Voronoï cell complexes attached to the modular groups and . From this we deduce the rational cohomology of those groups and some information about , when and 7.
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Let X be an arithmetic surface, and let L be a line bundle on X. Choose a metric h on the lattice Λ of sections of L over X. When the degree of the generic fiber of X is large enough, we get lower bounds for the successive minima of (Λ, h) in terms of the normalized height of X. Theproofusesan effective version (due to C. Voisin) of a theorem of Se...
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We consider a short sequence of hermitian vector bundles on some arithmetic variety. Assuming that this sequence is exact on the generic fiber we prove that the alternated sum of the arithmetic Chern characters of these bundles is the sum of two terms, namely the secondary Bott Chern character class of the sequence and its Chern character with supp...
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For N=5, 6 and 7, using the classification of perfect quadratic forms, we compute the homology of the Voronoi cell complexes attached to the modular groups SL_N(\Z) and GL_N(\Z). From this we deduce the rational cohomology of those groups. Comment: 17 pages, 7 tables and figures
Article
We associate weight complexes of (homological) motives, and hence Euler characteristics in the Grothendieck group of motives, to arithmetic varieties and Deligne–Mumford stacks; this extends the results in the paper [H. Gillet, C. Soulé, Descent, motives and K-theory, J. Reine Angew. Math. 478 (1996) 127–176], where a similar result was proved for...
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We prove an analogue in Arakelov geometry of the Grothendieck-Riemann-Roch theorem.
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This letter reports complete sets of two-fold symmetries between partitions of the universal genetic code. By substituting bases at each position of the codons according to a fixed rule, it happens that properties of the degeneracy pattern or of tRNA aminoacylation specificity are exchanged.
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We consider a dynamical system, described by a system of ordinary differential equations, and the associated interaction graphs, which are defined using the matrix of signs of the Jacobian matrix. After stating a few conjectures about the role of circuits in these graphs, we prove two new results relating them to the dynamic behaviour of the system...
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We consider heights of horizontal irreducible divisors on an arithmetic surface with respect to some hermitian line bundle. We obtain both lower and upper bounds for these heights. The results are different and sometimes stronger that those of S.Zhang on the same question. The case of the relative dualizing sheaf with the Arakelov metric is made es...
Article
We consider some mathematical issues raised by the modelling of gene networks. The expression of genes is governed by a complex set of regulations, which is often described symbolically by interaction graphs. These are finite oriented graphs where vertices are the genes involved in the biological system of interest and arrows describe their interac...
Article
Atiyah and Hirzebruch gave examples of even degree torsion classes in the singular cohomology of a smooth complex projective manifold, which are not Poincaré dual to an algebraic cycle. We notice that the order of these classes must be small compared to the dimension of the manifold. However, building upon a construction of Kollár, one can provide...
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On a given arithmetic surface, inspired by work of Miyaoka, we consider vector bundles which are extensions of a line bundle by another one. We give sufficient conditions for their restriction to the generic fiber to be semi-stable. We then apply the arithmetic analog of Bogomolov inequality in Arakelov theory, and deduce from it a lower bound for...
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Arithmetic groups are groups of matrices with integral entries. We shall first discuss their origin in number theory (Gauss, Minkowski) and their role in the "reduction theory of quadratic forms". Then we shall describe these groups by generators and relations. The next topic will be: are all subgroups of finite index given by congruence conditions...
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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...
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We propose a definition of varieties over the field with one element. These have extensions of scalars to the ring of integers which are varieties in the usual sense. We show that toric varieties can be defined over the field with one element. We also discuss zeta functions for such objects. We give a motivic interpretation of the image of the J-ho...
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Let m be an integer bigger than one, A a ring of algebraic integers, F its fraction field, and Km(A) the m-th Quillen K-group of A. We give a (huge) explicit bound for the order of the torsion subgroup of Km(A) (up to small primes), in terms of m, the degree of F over Q, and its absolute discriminant.
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Pour N=5 et N=6, nous calculons le complexe cellulaire défini par Voronoï à partir des formes quadratiques réelles de dimension N. Nous en déduisons l'homologie de à coefficients triviaux, à de petits nombres premiers près. Nous montrons aussi que et que n'a que de la 3-torsion. Pour citer cet article : P. Elbaz-Vincent et al., C. R. Acad. Sci. Par...
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For N=5 and N=6, we compute the Voronoi cell complex attached to real N-dimensional quadratic forms, and we obtain the homology of GL_N(Z) with trivial coefficients, up to small primes. We also prove that K_5(Z) = Z and K_6(Z) has only 3-torsion.
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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 i...
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Let 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 , 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 groupis a finite group of order a power of 2.This result was proved 1...
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We develop a formalism of direct images for metrized vector bundles in the context of the non-archimedean Arakelov theory introduced in our previous joint work with S. Bloch, and we prove a Riemann-Roch-Grothendieck theorem for this direct image.
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Let p be an odd prime, n an odd positive integer and C the p-Sylow subgroup the class group of the p-cyclotomic extension of the rationals. When log(p) is bigger than n**(224n**4), we prove that the eigenspace on C attached to the (p-n)-th power of the Teichmuller character is trivial. The proof uses the K-theory of the integers and the Voronoi red...
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Let $E$ be a holomorphic vector bundle on a compact K\"ahler manifold $X$. If we fix a metric $h$ on $E$, we get a Laplace operator $\Delta$ acting upon smooth sections of $E$ over $X$. Using the zeta function of $\Delta$, one defines its regularized determinant $det'(\Delta)$. We conjectured elsewhere that, when $h$ varies, this determinant $det'(...
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To an arbitrary variety over a field of characteristic zero, we associate a complex of Chow motives, which is, up to homotopy, unique and bounded. We deduce that any variety has a natural Euler characteristic in the Grothendieck group of Chow motives. We show that the cohomology with integer coefficients of any singular variety over the complex num...
Article
Using arithmetic intersection theory, a theory of heights for projective varieties over rings of algebraic integers is developed. These heights are generalizations of those considered by Weil, Schmidt, Nesterenko, Philippon, and Faltings. Several of their properties are proved, including lower bounds and an arithmetic Bézout theorem for the height...
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The purpose of this note is to remark that Theorem 3.7 in [1], when combined with the work of Bismut and Freed [2], leads, in the algebraic case, to an improvement of both results concerning the holonomy of determinant line bundles.
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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.
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In this paper, we prove that in the case of holomorphic locally Khler fibrations, the analytic and algebraic geometry constructions of determinant bundles for direct images coincide. We calculate the curvature of the holomorphic Hermitian connection for the Quillen metric on the determinant bundle. We study the behavior of the Quillen metric under...
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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 form...
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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 Quillen's superconnections. We also give an axiomatic characterization of these classes. These results will be used in [BGS2] and [BGS3] to study the dete...
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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...
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C'est pour étendre le théorème de Riemann-Roch à un morphisme projectif arbitraire que Grothendieck a introduit le groupe K(X) (noté aujourd'hui K 0 ( X )), construit à l'aide des -modules localement libres sur un schéma X [ 14 ]. La somme directe et le produit tensoriel de modules font de K 0 ( X ) un anneau, et les opérations de puissances extéri...
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CH' (X)le groupe des cycles de codimension isurX modulo l'equivalence rationnelle (i.e. modulo le groupe engendre par les diviseurs des fonctions sur les sous-varietes de codimension i -1de X). Le groupe CH °(X)est Z, et CH '(X)n'est autre que le groupe de Picard Pic(X) . Comme pour X/k projective le foncteurest representable, on peut, en utilisant...

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