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

Abstract

Let XXbe a smooth proper variety over the quotient field of a Henselian discrete valuation ring with algebraically closed residue field of characteristic pp. We show that for any coherent sheaf EEon XX, the index of XXdivides the Euler–Poincaré characteristicχ(X,E)\chi (X,E)ifp=0p=0orp>dim⁡(X)+1p>\dim (X)+1. If0>p≤dim⁡(X)+10>p\leq \dim (X)+1, the prime-to-pppart of the index of XXdividesχ(X,E)\chi (X,E). Combining this with the Hattori–Stong theorem yields an analogous result concerning the divisibility of the cobordism class of XXby the index of XX.As a corollary, rationally connected varieties over the maximal unramified extension of app-adic field possess a zero-cycle ofpp-power degree (a zero-cycle of degree 11ifp>dim⁡(X)+1p>\dim (X)+1). Whenp=0p=0, such statements also have implications for the possible multiplicities of singular fibers in degenerations of complex projective varieties.