Comptes Rendus
no. 4
Analytic geometry/Differential geometry
A Riemann–Roch–Grothendieck theorem for flat fibrations with complex fibers
[Un théorème de Riemann–Roch–Grothendieck pour une fibration plate de fibre complexe]

Présenté par : Jean-Michel Bismut

Yeping Zhang1
1 Département de mathématiques, bâtiment 425, faculté des sciences d'Orsay, Université Paris-Sud, 91405 Orsay cedex, France
Comptes Rendus. Mathématique, Volume 354 (2016) no. 4, pp. 401-406.

On considère une fibration propre plate de base réelle et de fibre complexe. On construit d'abord des classes caractéristiques impaires [5] associées qui généralisent des constructions de Bismut–Lott [5]. Puis on considère l'image directe d'un fibré vectoriel holomorphe dans la fibre, qui est un fibré vectoriel plat sur la base. On donne un théorème de Riemann–Roch–Grothendieck calculant les classes caractéristiques impaires de ce fibré plat.

We consider a proper flat fibration with real base and complex fibers. First we construct odd characteristic classes for such fibrations by a method that generalizes constructions of Bismut–Lott [5]. Then we consider the direct image of a fiberwise holomorphic vector bundle, which is a flat vector bundle on the base. We give a Riemann–Roch–Grothendieck theorem calculating the odd real characteristic classes of this flat vector bundle.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2016.01.011
Yeping Zhang 1

1 Département de mathématiques, bâtiment 425, faculté des sciences d'Orsay, Université Paris-Sud, 91405 Orsay cedex, France 
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Yeping Zhang. A Riemann–Roch–Grothendieck theorem for flat fibrations with complex fibers. Comptes Rendus. Mathématique, Volume 354 (2016) no. 4, pp. 401-406. doi : 10.1016/j.crma.2016.01.011. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2016.01.011/

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[5] J.-M. Bismut; J. Lott Flat vector bundles, direct images and higher real analytic torsion, J. Amer. Math. Soc., Volume 8 (1995) no. 2, pp. 291-363

[6] R.T. Seeley Complex powers of an elliptic operator, Proc. Sympos. Pure Math., Chicago, IL, 1966, Amer. Math. Soc., Providence, R.I. (1967), pp. 288-307

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