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.
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@article{CRMATH_2016__354_4_401_0, author = {Zhang, Yeping}, title = {A {Riemann{\textendash}Roch{\textendash}Grothendieck} theorem for flat fibrations with complex fibers}, journal = {Comptes Rendus. Math\'ematique}, pages = {401--406}, publisher = {Elsevier}, volume = {354}, number = {4}, year = {2016}, doi = {10.1016/j.crma.2016.01.011}, language = {en}, url = {http://archive.numdam.org/articles/10.1016/j.crma.2016.01.011/} }
TY - JOUR AU - Zhang, Yeping TI - A Riemann–Roch–Grothendieck theorem for flat fibrations with complex fibers JO - Comptes Rendus. Mathématique PY - 2016 SP - 401 EP - 406 VL - 354 IS - 4 PB - Elsevier UR - http://archive.numdam.org/articles/10.1016/j.crma.2016.01.011/ DO - 10.1016/j.crma.2016.01.011 LA - en ID - CRMATH_2016__354_4_401_0 ER -
%0 Journal Article %A Zhang, Yeping %T A Riemann–Roch–Grothendieck theorem for flat fibrations with complex fibers %J Comptes Rendus. Mathématique %D 2016 %P 401-406 %V 354 %N 4 %I Elsevier %U http://archive.numdam.org/articles/10.1016/j.crma.2016.01.011/ %R 10.1016/j.crma.2016.01.011 %G en %F CRMATH_2016__354_4_401_0
Zhang, Yeping. A Riemann–Roch–Grothendieck theorem for flat fibrations with complex fibers. Comptes Rendus. Mathématique, Tome 354 (2016) no. 4, pp. 401-406. doi : 10.1016/j.crma.2016.01.011. http://archive.numdam.org/articles/10.1016/j.crma.2016.01.011/
[1] The index of elliptic operators, IV, Ann. of Math. (2), Volume 93 (1971), pp. 119-138
[2] The Atiyah–Singer index theorem for families of Dirac operators: two heat equation proofs, Invent. Math., Volume 83 (1986) no. 1, pp. 91-151
[3] Analytic torsion and holomorphic determinant bundles, III: quillen metrics on holomorphic determinants, Commun. Math. Phys., Volume 115 (1988) no. 2, pp. 301-351
[4] Higher analytic torsion forms for direct images and anomaly formulas, J. Algebraic Geom., Volume 1 (1992) no. 4, pp. 647-684
[5] Flat vector bundles, direct images and higher real analytic torsion, J. Amer. Math. Soc., Volume 8 (1995) no. 2, pp. 291-363
[6] 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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