Singular Bott-Chern classes and the arithmetic Grothendieck-Riemann-Roch theorem for closed immersions

TL;DR

This paper develops an axiomatic framework for singular Bott-Chern classes, classifies all such theories, and proves an extended arithmetic Grothendieck-Riemann-Roch theorem for closed immersions, with implications for arithmetic geometry.

Contribution

It provides an axiomatic classification of singular Bott-Chern classes and extends the arithmetic Grothendieck-Riemann-Roch theorem to closed immersions and projective morphisms.

Findings

Classified all theories of singular Bott-Chern classes.

Proved the arithmetic Grothendieck-Riemann-Roch theorem for closed immersions.

Established a Poincaré lemma for currents with fixed wave front set.

Abstract

We study the singular Bott-Chern classes introduced by Bismut, Gillet and Soule. 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 Soule 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 Soule 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 projective morphisms. As a byproduct of this study we obtain two results of independent interest. First, we prove a Poincare 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.