Index of varieties over Henselian fields and Euler characteristic of coherent sheaves

TL;DR

This paper establishes divisibility relations between the index of a smooth proper variety over Henselian fields and the Euler characteristic of coherent sheaves, with implications for zero-cycles and degenerations.

Contribution

It proves divisibility results linking the index and Euler characteristic over Henselian fields, extending to cobordism classes and zero-cycle existence.

Findings

The index divides the Euler characteristic when p=0 or p>dim(X)+1.

The prime-to-p part of the index divides the Euler characteristic for 0<p p dim(X)+1.

Rationally connected varieties over unramified p-adic extensions have zero-cycles of p-power degree.

Abstract

Let X be a smooth proper variety over the quotient field of a Henselian discrete valuation ring with algebraically closed residue field of characteristic p. We show that for any coherent sheaf E on X, the index of X divides the Euler-Poincar\'e characteristic \chi(X,E) if p=0 or p>dim(X)+1. If 0<p\leq dim(X)+1, the prime-to-p part of the index of X divides \chi(X,E). Combining this with the Hattori-Stong theorem yields an analogous result concerning the divisibility of the cobordism class of X by the index of X. As a corollary, rationally connected varieties over the maximal unramified extension of a p-adic field possess a zero-cycle of p-power degree (a zero-cycle of degree 1 if p>dim(X)+1). When p=0, such statements also have implications for the possible multiplicities of singular fibers in degenerations of complex projective varieties.