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

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-Poincaré characteristic χ(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 χ(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.

20 pages; final version

References in corpus (3)

• Algebraic cobordism
• Desingularization of quasi-excellent schemes in characteristic zero
• The index of an algebraic variety

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• The monodromy property for K3 surfaces allowing a triple-point-free model
• Examples of varieties with index one on $C_1$-fields
• Connective K-theory and Adams operations
• Étale degree map and 0-cycles
• The specialization index of a variety over a discretely valued field