Singular Bott-Chern classes and the arithmetic Grothendieck Riemann Roch theorem for closed immersions
Documenta mathematica, Tome 15 (2010), pp. 73-176.

Voir la notice de l'article dans Electronic Library of Mathematics

Summary: We study the singular Bott-Chern classes introduced by Bismut, Gillet and Soulé. Singular Bott-Chern classes are the main ingredient to define direct images for closed immersions in arithmetic $K$-theory. In this paper we give an axiomatic definition of a theory of singular Bott-Chern classes, study their properties, and classify all possible theories of this kind. We identify the theory defined by Bismut, Gillet and Soulé as the only one that satisfies the additional condition of being homogeneous. We include a proof of the arithmetic Grothendieck-Riemann-Roch theorem for closed immersions that generalizes a result of Bismut, Gillet and Soulé and was already proved by Zha. This result can be combined with the arithmetic Grothendieck-Riemann-Roch theorem for submersions to extend this theorem to arbitrary projective morphisms. As a byproduct of this study we obtain two results of independent interest. First, we prove a Poincaré lemma for the complex of currents with fixed wave front set, and second we prove that certain direct images of Bott-Chern classes are closed.
Classification : 14G40, 32U40
Mots-clés : Arakelov geometry, closed immersions, Bott-Chern classes, arithmetic Riemann-Roch theorem, currents, wave front sets 
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     author = {Burgos Gil, Jos\'e I. and Litcanu, Razvan},
     title = {Singular {Bott-Chern} classes and the arithmetic {Grothendieck} {Riemann} {Roch} theorem for closed immersions},
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Burgos Gil, José I.; Litcanu, Razvan. Singular Bott-Chern classes and the arithmetic Grothendieck Riemann Roch theorem for closed immersions. Documenta mathematica, Tome 15 (2010), pp. 73-176. https://geodesic-test.mathdoc.fr/item/DOCMA_2010__15__a31/