Néron-Severi groups under specialization. (English) Zbl 1248.14011

Maulik, Davesh; Poonen, Bjorn

Duke Math. J. 161, No. 11, 2167-2206 (2012).

In this remarkable article, the authors provide a \(p\)-adic approach to the study of the specialization properties of the Néron-Severi groups: they prove that in a family of varieties with good reduction at \(p\), the locus on the base where the Picard number jumps is \(p\)-adically nowhere dense. This allows to deduce the following result: for \(k\) an algebraically closed field of characteristic zero, \(B\) an irreducible variety over \(k\) and \(X\to B\) a smooth proper morphism, there exists a \(k\)-point of \(B\), such that the Picard number of the special fibre \(X_b\) is the same as the Picard number of the geometric generic fibre. This theorem was first established by Y. André [“Pour une théorie inconditionnelle des motifs”, Publ. Math., Inst. Hautes Étud. Sci. 83, 5–49 (1996; Zbl 0874.14010)] in the context of “motivated cycles”; the approach of Maulik and Poonen is completely different.
Let \(K\) be a field complete with respect to a dicrete valuation and let \(\mathcal O_K\) be its valuation ring, let \(C\) be the completion of an algebraic closure of \(K\) and let \(\mathcal O_C\) be its valuation ring. Let \(B\) be an irreducible separated \(\mathcal O_K\)-scheme of finite type. The main result of the article states that for \(f:X\to B\) a smooth proper family, the locus of \(b\in B(\mathcal O_C)\), where the Picard number of the \(C\)-variety \(X_b\) over \(b\) is strictly greater than the Picard number of the geometric generic fibre of \(f\), is nowhere dense in \(B(\mathcal O_C)\) in the analytic topology. Using crystalline methods, the authors give a local description of the locus where the Picard number jumps as a union of zeros of some power series. The main theorem is then follows using a subtle properties of \(p\)-adic power series.
As a consequence of their main result, the authors give an application for the study of the endomorphisms of abelian varieties, and for proper families of projective varieties. The detailed comparison with André’s method is also provided.
Assuming a \(p\)-adic version of the variational Hodge conjecture, the authors generalise their results for the higher dimensional cycles. The methods of the article can also be extended to the semistable case (see [G. Yamashita, “The \(p\)-adic Lefschetz \((1,1)\) theorem in the semistable case, and the Picard number jumping locus”, Math. Res. Lett. 18, No. 1, 107–124 (2011; Zbl 1238.14005)]).
The article is written in a beautiful way and the background material is also provided.

Reviewer: Alena Pirutka (Strasbourg)

MSC:

14C25	Algebraic cycles
14C22	Picard groups
14D05	Structure of families (Picard-Lefschetz, monodromy, etc.)
14F30	\(p\)-adic cohomology, crystalline cohomology

Keywords:

Néron-Severi groups; specialization maps; \(p\)-adic methods; crystalline cohomology

Citations:

Zbl 0874.14010; Zbl 1238.14005

References:

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