Hasse-Witt invariants of the reductions of certain irreducible symplectic varieties
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 an appropriate finite extension, one can obtain a similar statement with \(\Sigma\) of density 1. This result is parallel to earlier results by \textit{A.~Ogus} [\textit{P. Deligne} et al., Hodge cycles, motives, and Shimura varieties. Lecture Notes in Mathematics. 900. Berlin-Heidelberg-New York: Springer-Verlag. (1982; Zbl 0465.00010)] for abelian surfaces and by \textit{K. Joshi} and \textit{C. S. Rajan} [Int. Math. Res. Not. 2003, No. 2, 109--121 (2003; Zbl 1074.14019)]; it is better to look at \url{arxiv:math/0110070} rather than at the printed version] and \textit{F. A. Bogomolov} and \textit{Y. G. Zarhin} [Cent. Eur. J. Math. 7, No. 2, 206--213 (2009; Zbl 1178.14039)] for \(K3\) surfaces.
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