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

[Une formule de Riemann-Roch-Hirzebruch pour les traces d'opérateurs différentiels]

Engeli, Markus ; Felder, Giovanni

Annales scientifiques de l'École Normale Supérieure, Tome 41 (2008), p. 623-655 / Harvested from Numdam

Résumé

Soit $D$ un opérateur différentiel holomorphe opérant sur les sections d’un fibré vectoriel holomorphe sur une variété complexe de dimension $n$. Nous démontrons une formule, conjecturée par Feigin et Shoikhet, donnant le nombre de Lefschetz de $D$ comme intégrale d’une forme différentielle sur la variété. La classe de cette forme différentielle est obtenue, via la géométrie différentielle formelle du générateur canonique de la cohomologie de Hochschild $HH^{2n}(\mathcal {D}_n,\mathcal {D}_n^*)$ de l’algèbre des opérateurs différentiels sur un entourage formel d’un point. Si $D$ est l’identité, la formule se réduit à la formule de Riemann-Roch-Hirzebruch.

Abstract

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 $HH^{2n}(\mathcal {D}_n,\mathcal {D}_n^*)$ of the algebra of differential operators on a formal neighbourhood of a point. If $D$ is the identity, the formula reduces to the Riemann-Roch-Hirzebruch formula.

Détail

Publié le : 2008-01-01

MR 2489635 | Zbl 1163.32009 | 1 citation dans Numdam

DOI : https://doi.org/10.24033/asens.2077

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