Comptes Rendus
Géométrie différentielle
Laplacien hypoelliptique et cohomologie de Bott–Chern
[Hypoelliptic Laplacian and Bott–Chern cohomology]
Comptes Rendus. Mathématique, Volume 349 (2011) no. 1-2, pp. 75-80.

Let p:MS be a proper submersion of complex manifolds, and let F be a holomorphic vector bundle on M. When RpF 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.

Soit p:MS une submersion propre de variétés complexes, soit F un fibré vectoriel holomorphe sur M. Quand RpF 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.

Received:
Published online:
DOI: 10.1016/j.crma.2010.12.003

Jean-Michel Bismut 1

1 Département de mathématique, université Paris-Sud, bâtiment 425, 91405 Orsay cedex, France
@article{CRMATH_2011__349_1-2_75_0,
     author = {Jean-Michel Bismut},
     title = {Laplacien hypoelliptique et cohomologie de {Bott{\textendash}Chern}},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {75--80},
     publisher = {Elsevier},
     volume = {349},
     number = {1-2},
     year = {2011},
     doi = {10.1016/j.crma.2010.12.003},
     language = {fr},
}
TY  - JOUR
AU  - Jean-Michel Bismut
TI  - Laplacien hypoelliptique et cohomologie de Bott–Chern
JO  - Comptes Rendus. Mathématique
PY  - 2011
SP  - 75
EP  - 80
VL  - 349
IS  - 1-2
PB  - Elsevier
DO  - 10.1016/j.crma.2010.12.003
LA  - fr
ID  - CRMATH_2011__349_1-2_75_0
ER  - 
%0 Journal Article
%A Jean-Michel Bismut
%T Laplacien hypoelliptique et cohomologie de Bott–Chern
%J Comptes Rendus. Mathématique
%D 2011
%P 75-80
%V 349
%N 1-2
%I Elsevier
%R 10.1016/j.crma.2010.12.003
%G fr
%F CRMATH_2011__349_1-2_75_0
Jean-Michel Bismut. Laplacien hypoelliptique et cohomologie de Bott–Chern. Comptes Rendus. Mathématique, Volume 349 (2011) no. 1-2, pp. 75-80. doi : 10.1016/j.crma.2010.12.003. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2010.12.003/

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

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

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

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

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

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

[11] 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

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

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

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

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

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

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

Cited by Sources:

Comments - Policy