Index of varieties over Henselian fields and Euler characteristic of coherent sheaves
Résumé
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 $\chi(X,E)$ 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.