The flat Grothendieck-Riemann-Roch theorem without adiabatic techniques

In this paper we give a simplified proof of the flat Grothendieck-Riemann-Roch theorem. The proof makes use of the local family index theorem and basic computations of the Chern-Simons form. In particular, it does not involve any adiabatic limit computation of the reduced eta-invariant.Comment: 21 pages. Comments are welcome. Final version. To appear in Journal of Geometry and Physic

Differential K-theory. A survey

Generalized differential cohomology theories, in particular differential K-theory (often called "smooth K-theory"), are becoming an important tool in differential geometry and in mathematical physics. In this survey, we describe the developments of the recent decades in this area. In particular, we discuss axiomatic characterizations of differential K-theory (and that these uniquely characterize differential K-theory). We describe several explicit constructions, based on vector bundles, on families of differential operators, or using homotopy theory and classifying spaces. We explain the most important properties, in particular about the multiplicative structure and push-forward maps and will state versions of the Riemann-Roch theorem and of Atiyah-Singer family index theorem for differential K-theory.Comment: 50 pages, report based in particular on work done sponsored the DFG SSP "Globale Differentialgeometrie". v2: final version (only typos corrected), to appear in C. B\"ar et al. (eds.), Global Differential Geometry, Springer Proceedings in Mathematics 17, Springer-Verlag Berlin Heidelberg 201

Noncommutative Geometry

These reports contain an account of 2015’s meeting on noncommutative geometry. Noncommutative geometry has developed itself over the years to a completely new branch of mathematics shedding light on many other areas as number theory, differential geometry and operator algebras. A connection that was highlighted in particular in this meeting was the connection with the theory of $II_1$-factors and geometric group theory

Limites adiabatiques, fibrations holomorphes plates et théorème de R.R.G.

This thesis consists of two parts. The first part is an article written jointly with Martin Puchol and Jialin Zhu, the second part is a series of results obtained by myself in connection with the Riemann-Roch-Grothendieck theorem for flat vector bundles. In the first part, we give an analytic approach to the behavior of classical Ray-Singer analytic torsion in de Rham theory when a manifold is separated along a hypersurface. More precisely, we give a formula relating the analytic torsion of the full manifold, and the analytic torsion associated with relative or absolute boundary conditions along the hypersurface. In the second part of this thesis, we refine the results of Bismut-Lott on direct images of flat vector bundles to the case where the considered flat vector bundle is itself the fiberwise holomorphic cohomology of a vector bundle along a flat fibration by complex manifolds. In this context, we give a formula of Riemann-Roch-Grothendieck in which the Todd class of the relative holomorphic tangent bundle appears explicitly. By replacing cohomology classes by explicit differential forms in Chern-Weil theory, we extend the constructions of Bismut-Lott in this context.Cette thèse est faite de deux parties. La première partie est un article rédigé conjointementavec Martin Puchol et Jialin Zhu. La deuxième partie est une série de résultats obtenus par moi-même liés au théorème de Riemann-Roch-Grothendieck pour les fibrés vectoriels plats. Dans la première partie, nous donnons une preuve analytique d'un résultat décrivant le comportement de la torsion analytique en théorie de de Rham lorsque la variété considérée est séparée en deux par une hypersurface. Plus précisément, nous donnons une formule liant la torsion analytique de la variété entière aux torsions analytiques associées aux variétés à bord avec des conditions limites relative ou absolue le long de l'hypersurface. Dans la deuxième partie de cette thèse, nous raffinons les résultats de Bismut-Lott pour les images directes des fibrés vectoriels plats au cas où le fibré vectoriel plat en question est lui-même la cohomologie holomorphe d'un fibré vectoriel le long d'une fibration plate à fibres complexes. Dans ce contexte, nous donnons une formule de Riemann-Roch-Grothendieck dans laquelle la classe de Todd du fibré tangent relatif apparaît explicitement. En remplaçant les classes de cohomologie par des formes explicites qui les représentent en théorie de Chern-Weil, nous généralisons ainsi des constructions de Bismut-Lott

Coherent sheaves, superconnections, and RRG

Given a compact complex manifold, the purpose of this paper is to construct the Chern character for coherent sheaves with values in Bott-Chern cohomology, and to prove a corresponding Riemann-Roch-Grothendieck formula. Our paper is based on a fundamental construction of Block.Comment: 161 pages, 1 figure. In version 2, references to earlier work have been adde