no. 1-2
Géométrie différentielle
Laplacien hypoelliptique et cohomologie de Bott–Chern

Présenté par : Jean-Michel Bismut

Bismut, Jean-Michel1
1 Département de mathématique, université Paris-Sud, bâtiment 425, 91405 Orsay cedex, France
Comptes Rendus. Mathématique, Tome 349 (2011) no. 1-2, pp. 75-80.

Soit p:M→S une submersion propre de variétés complexes, soit F un fibré vectoriel holomorphe sur M. Quand R⋅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 ∂¯∂ 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→S be a proper submersion of complex manifolds, and let F be a holomorphic vector bundle on M. When R⋅p⁎F is locally free, we establish a Riemann–Roch–Grothendieck theorem in Bott–Chern cohomology. When M is equipped with a ∂¯∂-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.

Reçu le :
Publié le :
DOI : 10.1016/j.crma.2010.12.003
Bismut, Jean-Michel 1

1 Département de mathématique, université Paris-Sud, bâtiment 425, 91405 Orsay cedex, France 
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Bismut, Jean-Michel. Laplacien hypoelliptique et cohomologie de Bott–Chern. Comptes Rendus. Mathématique, Tome 349 (2011) no. 1-2, pp. 75-80. doi : 10.1016/j.crma.2010.12.003. http://archive.numdam.org/articles/10.1016/j.crma.2010.12.003/

[1] Atiyah, M.F.; Singer, I.M. The index of elliptic operators. IV, Ann. of Math. (2), Volume 93 (1971), pp. 119-138

[2] Berline, N.; Getzler, E.; Vergne, M. Heat Kernels and Dirac Operators, Grundl. Math. Wiss., Band 298, Springer-Verlag, Berlin, 1992

[3] Bismut, J.-M. The Atiyah–Singer index theorem for families of Dirac operators: two heat equation proofs, Invent. Math., Volume 83 (1986) no. 1, pp. 91-151

[4] Bismut, J.-M. A local index theorem for non-Kähler manifolds, Math. Ann., Volume 284 (1989) no. 4, pp. 681-699

[5] Bismut, J.-M. The hypoelliptic Laplacian on the cotangent bundle, J. Amer. Math. Soc., Volume 18 (2005) no. 2, pp. 379-476 (electronic)

[6] Bismut, J.-M. The hypoelliptic Dirac operator, Geometry and Dynamics of Groups and Spaces, Progr. Math., vol. 265, Birkhäuser, Basel, 2008, pp. 113-246

[7] J.-M. Bismut, Hypoelliptic Laplacian and Bott–Chern cohomology, preprint (Orsay), 2011.

[8] Bismut, J.-M.; Gillet, H.; Soulé, C. Analytic torsion and holomorphic determinant bundles. II. Direct images and Bott–Chern forms, Comm. Math. Phys., Volume 115 (1988) no. 1, pp. 79-126

[9] Bismut, J.-M.; Köhler, K. Higher analytic torsion forms for direct images and anomaly formulas, J. Algebraic Geom., Volume 1 (1992) no. 4, pp. 647-684

[10] Bismut, J.-M.; Lebeau, G. The Hypoelliptic Laplacian and Ray–Singer Metrics, Annals of Mathematics Studies, vol. 167, Princeton University Press, Princeton, NJ, 2008

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

[12] Bott, R.; Chern, S.S. Hermitian vector bundles and the equidistribution of the zeroes of their holomorphic sections, Acta Math., Volume 114 (1965), pp. 71-112

[13] Demailly, J.-P. Complex analytic and differential geometry, 2009 http://www-fourier.ujf-grenoble.fr/~demailly/manuscripts/agbook.pdf (OpenContent Book)

[14] Grauert, H. Ein Theorem der analytischen Garbentheorie und die Modulräume komplexer Strukturen, Inst. Hautes Études Sci. Publ. Math., Volume 5 (1960), p. 64

[15] Grivaux, J. Chern classes in Deligne cohomology for coherent analytic sheaves, Math. Ann., Volume 347 (2010) no. 2, pp. 249-284

[16] Quillen, D. Superconnections and the Chern character, Topology, Volume 24 (1985) no. 1, pp. 89-95

[17] Schweitzer, M. Autour de la cohomologie de Bott–Chern, 2007 | arXiv

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