Hasse-Witt invariants of the reductions of certain irreducible symplectic varieties
type: Item
instance of: scholarly article
title: Hasse-Witt invariants of the reductions of certain irreducible symplectic varieties
zbMATH Open document ID: 1299.14025
author: Stéphane Druel
published in: Michigan Mathematical Journal
publication date: 2012-10-31T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0110070
full work available at URL: https://projecteuclid.org/euclid.mmj/1347040262
review text: The main object of the paper under review is the Hasse-Witt invariant of an algebraic variety \(X\) defined over a perfect field of characteristic \(p>0\). It is defined as follows. If \(V\) is a finite-dimensional vector \(k\)-space, \(\sigma\) is the Frobenius automorphism of \(k\) and \(u\) is a \(\sigma\)-linear endomorphism of \(V\), one has a decomposition of \(V\) into a direct sum \(V=V_{\mathrm{ss}}\oplus V_{\mathrm{nil}}\) of \(u\)-invariant spaces, where \(V_{\mathrm{ss}}=\bigcap_{i\geq 0}u^i(V)\), \(V_{\mathrm{nil}}=\bigcup_{i\geq 0}\ker(u^i)\). The dimension of \(V_{\mathrm{ss}}\) is then called the stable rank of \(u\). If now \(X\) is a proper algebraic \(k\)-variety of dimension \(d\) and \(F: X\to X\) is the absolute Frobenius morphism, then the Hasse-Witt invariant of \(X\) is, by definition, the stable rank of \(F\) acting on \(H^d(X,O_X)\). The paper is motivated by the following: Conjecture. If \(K\) is a number field, \(X\) is a smooth projective \(K\)-variety, \(\mathcal X\) is an integer model of \(X\) defined over the ring of integers \(O_K\) of \(K\), \(\mathcal V\) is a finite set of non-archimedean places of \(K\) such as the model is smooth outside \(\mathcal V\), then there exists a positive density set \(\Sigma\) of places of \(K\) such that for every \(v\in\Sigma\setminus \mathcal V\) the Hasse-Witt invariant of the reduction \(\mathcal X_v\) is maximal. The author's focus is on the case where \(X\) is an irreducible symplectic \(K\)-variety. This means that \(X\) is geometrically integral, carries a closed 2-form which is everywhere nondegenerate, and for the Hodge numbers of \(X\) one has \(h^{0,1}=0\), \(h^{2,0}=1\). The main result of the paper states that if \(X\) is polarizable and \(b_2(X\otimes \mathbb C)>3\), then there exists a positive density set \(\Sigma\) of places of \(K\) such that for every \(v\in\Sigma\setminus \mathcal V\) the Hasse-Witt invariant of \(\mathcal X_v\) is nonzero. Moreover, after replacing \(K\) with …
Mathematics Subject Classification ID: 14G25
Mathematics Subject Classification ID: 14C25
Mathematics Subject Classification ID: 14G15
Mathematics Subject Classification ID: 11G25
Mathematics Subject Classification ID: 11G35
zbMATH DE Number: 6100775
zbMATH Keywords: symplectic variety
zbMATH Keywords: Hasse--Witt invariant
reviewed by: Boris Kunyavskiĭ
MaRDI profile type: Publication
OpenAlex ID: W2026606501
arXiv ID: math/0110070
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DOI: 10.1307/MMJ/1347040262
description: scientific article