Singular Bott-Chern classes and the arithmetic Grothendieck-Riemann-Roch theorem for closed immersions - Mathematics > Algebraic GeometryReport as inadecuate




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Abstract: We study the singular Bott-Chern classes introduced by Bismut, Gillet andSoule. Singular Bott-Chern classes are the main ingredient to define directimages for closed immersions in arithmetic K-theory. In this paper we give anaxiomatic definition of a theory of singular Bott-Chern classes, study theirproperties, and classify all possible theories of this kind. We identify thetheory defined by Bismut, Gillet and Soule as the only one that satisfies theadditional condition of being homogeneous. We include a proof of the arithmeticGrothendieck-Riemann-Roch theorem for closed immersions that generalizes aresult of Bismut, Gillet and Soule and was already proved by Zha. This resultcan be combined with the arithmetic Grothendieck-Riemann-Roch theorem forsubmersions to extend this theorem to projective morphisms. As a byproduct ofthis study we obtain two results of independent interest. First, we prove aPoincare lemma for the complex of currents with fixed wave front set, andsecond we prove that certain direct images of Bott-Chern classes are closed.



Author: J. I. Burgos Gil, R. Litcanu

Source: https://arxiv.org/







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