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

Esnault, Hélène; Levine, Marc; Wittenberg, Olivier

doi:10.1090/jag/639

Authors: Hélène Esnault, Marc Levine and Olivier Wittenberg
Journal: J. Algebraic Geom. 24 (2015), 693-718
DOI: https://doi.org/10.1090/jag/639
Published electronically: June 18, 2015
MathSciNet review: 3383601
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Abstract | References | Additional Information

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 $\chi (X,E)$ 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.

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Additional Information

Hélène Esnault
Affiliation: Universität Duisburg-Essen, Fakultät Mathematik, Campus Essen, 45117 Essen, Germany
Address at time of publication: FU Berlin, FB Mathematik und Informatik, Arnimalee 3, 14195 Berlin, Germany
MR Author ID: 64210
Email: esnault@math.fu-berlin.de

Marc Levine
Affiliation: Universität Duisburg-Essen, Fakultät Mathematik, Campus Essen, 45127 Essen, Germany
MR Author ID: 113315
Email: marc.levine@uni-due.de

Olivier Wittenberg
Affiliation: Département de mathématiques et applications, École normale supérieure, 45 rue d’Ulm, 75230 Paris Cedex 05, France
MR Author ID: 729226
Email: wittenberg@dma.ens.fr

Received by editor(s): September 26, 2012
Received by editor(s) in revised form: January 9, 2013
Published electronically: June 18, 2015
Additional Notes: The first author was supported by the Einstein Foundation, the ERC Advanced Grant 226257 and the Chaire d’Excellence 2011 de la Fondation Sciences Mathématiques de Paris. The second author was supported by the Alexander von Humboldt Foundation
Article copyright: © Copyright 2015 University Press, Inc.
The copyright for this article reverts to public domain 28 years after publication.