On the arithmetic Chern character

TL;DR

This paper investigates the properties of the arithmetic Chern character in exact sequences of hermitian vector bundles on arithmetic varieties, introducing new formulas involving secondary classes and supports, with applications to arithmetic surfaces.

Contribution

It provides a new formula for the arithmetic Chern character of exact sequences, incorporating secondary Bott Chern classes and support terms, with explicit computations for arithmetic surfaces.

Findings

Derived a formula relating the sum of arithmetic Chern characters to secondary classes.

Computed these classes explicitly in the context of arithmetic surfaces.

Applied results to a Kodaira vanishing theorem for arithmetic surfaces.

Abstract

We consider a short sequence of hermitian vector bundles on some arithmetic variety. Assuming that this sequence is exact on the generic fiber we prove that the alternated sum of the arithmetic Chern characters of these bundles is the sum of two terms, namely the secondary Bott Chern character class of the sequence and its Chern character with supports on the finite fibers. Next, we compute these classes in the situation encountered by the second author when proving a "Kodaira vanishing theorem" for arithmetic surfaces.