Laplacien hypoelliptique et cohomologie de Bott–Chern

Bismut, Jean-Michel

Comptes Rendus. Mathématique, Tome 349 (2011) no. 1-2, pp. 75-80.

Soit $p:M\to S$ une submersion propre de variétés complexes, soit F un fibré vectoriel holomorphe sur M. Quand $R^{\cdot }{p}_{⁎}F$ est localement libre, on établit un théorème de Riemann–Roch–Grothendieck en cohomologie de Bott–Chern. Quand M est munie d'une $(1,1)$ forme $\overline{\partial }\partial$ fermée induisant une métrique Hermitienne le long des fibre, la preuve résulte d'une modification convenable des superconnexions elliptiques. Dans le cas général, on construit une version exotique des superconnexions hypoelliptiques que nous avons introduites dans des travaux antérieurs.

Let $p:M\to S$ be a proper submersion of complex manifolds, and let F be a holomorphic vector bundle on M. When $R^{\cdot }{p}_{⁎}F$ is locally free, we establish a Riemann–Roch–Grothendieck theorem in Bott–Chern cohomology. When M is equipped with a $\overline{\partial }\partial$-closed $(1,1)$ form inducing a Hermitian metric along the fibres, the proof is obtained by using elliptic superconnections. In the general case, we construct an exotic version of the hypoelliptic superconnections which we introduced in previous work.

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