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A Riemann-Roch-Hirzebruch formula for traces of differential operators

Markus Engeli, Giovanni Felder (2008)

Annales scientifiques de l'École Normale Supérieure

Let D be a holomorphic differential operator acting on sections of a holomorphic vector bundle on an n -dimensional compact complex manifold. We prove a formula, conjectured by Feigin and Shoikhet, giving the Lefschetz number of D as the integral over the manifold of a differential form. The class of this differential form is obtained via formal differential geometry from the canonical generator of the Hochschild cohomology H H 2 n ( 𝒟 n , 𝒟 n * ) of the algebra of differential operators on a formal neighbourhood of a...

Asymptotic behaviour of numerical invariants of algebraic varieties

F. L. Zak (2012)

Journal of the European Mathematical Society

We show that if the degree of a nonsingular projective variety is high enough, maximization of any of the most important numerical invariants, such as class, Betti number, and any of the Chern or middle Hodge numbers, leads to the same class of extremal varieties. Moreover, asymptotically (say, for varieties whose total Betti number is big enough) the ratio of any two of these invariants tends to a well-defined constant.

Formal geometric quantization

Paul-Émile Paradan (2009)

Annales de l’institut Fourier

Let K be a compact Lie group acting in a Hamiltonian way on a symplectic manifold ( M , Ω ) which is pre-quantized by a Kostant-Souriau line bundle. We suppose here that the moment map Φ is proper so that the reduced space M μ : = Φ - 1 ( K · μ ) / K is compact for all μ . Then, we can define the “formal geometric quantization” of M as 𝒬 K - ( M ) : = μ K ^ 𝒬 ( M μ ) V μ K . The aim of this article is to study the functorial properties of the assignment ( M , K ) 𝒬 K - ( M ) .

Generalized holomorphic analytic torsion

José Ignacio Burgos Gil, Gerard Freixas i Montplet, Răzvan Liţcanu (2014)

Journal of the European Mathematical Society

In this paper we extend the holomorphic analytic torsion classes of Bismut and Köhler to arbitrary projective morphisms between smooth algebraic complex varieties. To this end, we propose an axiomatic definition and give a classification of the theories of generalized holomorphic analytic torsion classes for projective morphisms. The extension of the holomorphic analytic torsion classes of Bismut and Köhler is obtained as the theory of generalized analytic torsion classes associated to R = 2 , R being...

Karoubi’s relative Chern character and Beilinson’s regulator

Georg Tamme (2012)

Annales scientifiques de l'École Normale Supérieure

We construct a variant of Karoubi’s relative Chern character for smooth varieties over 𝐂 and prove a comparison result with Beilinson’s regulator with values in Deligne-Beilinson cohomology. As a corollary we obtain a new proof of Burgos’ Theorem that for number fields Borel’s regulator is twice Beilinson’s regulator.

Proof of Nadel’s conjecture and direct image for relative K -theory

Alain Berthomieu (2002)

Bulletin de la Société Mathématique de France

A “relative” K -theory group for holomorphic or algebraic vector bundles on a compact or quasiprojective complex manifold is constructed, and Chern-Simons type characteristic classes are defined on this group in the spirit of Nadel. In the projective case, their coincidence with the Abel-Jacobi image of the Chern classes of the bundles is proved. Some applications to families of holomorphic bundles are given and two Riemann-Roch type theorems are proved for these classes.

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