Henri Gillet

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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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.

Parameterized Picard–Vessiot extensions and Atiyah extensions

Abstract Generalizing Atiyah extensions, we introduce and study differential abelian tensor categories over differential rings. By a differential ring, we mean a commutative ring with an action of a Lie ring by derivations. In particular, these derivations act on a differential category. A differential Tannakian theory is developed. The main application is to the Galois theory of linear differential equations with parameters. Namely, we show the existence of a parameterized Picard–Vessiot extension and, therefore, the Galois correspondence for many differential fields with, possibly, non-differentially closed fields of constants, that is, fields of functions of parameters. Other applications include a substantially simplified test for a system of linear differential equations with parameters to be isomonodromic, which will appear in a separate paper. This application is based on differential categories developed in the present paper, and not just differential algebraic groups and their representations.

DIFFERENTIAL ALGEBRA A SCHEME THEORY APPROACH

Two results in Differential Algebra, Kolchin’s irreducibility theorem, and a result on descent of projective varieties (due to Buium) are proved using methods of “modern” or “Grothendieck style” algebraic geometry.

Gersten′s conjecture for theK-theory with torsion coefficients of a discrete valuation ring

between Quillen K-groups induced by the obvious inclusion functors, are all zero. In [6] Quillen proved this conjecture if R is essentially of finite type over a field. In the case of discrete valuation rings, this conjecture reduces to: Conjecture. Let R be a discrete valuation ring with residue field k and field of fractions F. Then for all q z 0, the transfer map (which is defined since every k vector space is of finite projective dimension as an R module): K,(k) -+ K,(R) is zero. Equivalently, the map &(R) --) EC,(F) induced by the ring homomorphism R -P F is injective. A third equivalent formulation is that the boundary map:

Deligne homology and Abel-Jacobi maps

The purpose of this note is to announce a functorial description of the Abel-Jacobi homomorphisms of [7] by means of a theory of cycle classes taking values in Deligne homology, which is part of a Poincaré duality theory satisfying the axioms of [3] on the category of all schemes of finite type over C. One consequence of this approach is that the cycle class map is an edge homomorphism in the coniveau spectral sequence, and so conjectures on the structure of the Chow groups, such as Blochs conjecture [2], can be interpreted in terms of the vanishing (or not) of differentials in this spectral sequence. This formalism, because of its distinction between homology and cohomology, may also give a good framework for studying specialization questions. Deligne cohomology was originally defined by Deligne for X a proper smooth variety (or manifold) over C:

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.

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.