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

Abstract

Let  X X be a smooth proper variety over the quotient field of a Henselian discrete valuation ring with algebraically closed residue field of characteristic  p p . We show that for any coherent sheaf  E E on  X X , the index of  X X divides the Euler–Poincaré characteristic χ ( X , E ) \chi (X,E) if p = 0 p=0 or p > dim ⁡ ( X ) + 1 p>\dim (X)+1 . If 0 > p ≤ dim ⁡ ( X ) + 1 0>p\leq \dim (X)+1 , the prime-to- p p part of the index of  X X divides χ ( X , E ) \chi (X,E) . Combining this with the Hattori–Stong theorem yields an analogous result concerning the divisibility of the cobordism class of  X X by the index of  X X .

As a corollary, rationally connected varieties over the maximal unramified extension of a p p -adic field possess a zero-cycle of p p -power degree (a zero-cycle of degree  1 1 if p > dim ⁡ ( X ) + 1 p>\dim (X)+1 ). When p = 0 p=0 , such statements also have implications for the possible multiplicities of singular fibers in degenerations of complex projective varieties.