Duke Mathematical Journal

Néron–Severi groups under specialization

Davesh Maulik and Bjorn Poonen

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Abstract

André used Hodge-theoretic methods to show that in a smooth proper family XB of varieties over an algebraically closed field k of characteristic zero, there exists a closed fiber having the same Picard number as the geometric generic fiber, even if k is countable. We give a completely different approach to André’s theorem, which also proves the following refinement: 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. Our proof uses the “p-adic Lefschetz (1,1)-theorem” of Berthelot and Ogus, combined with an analysis of p-adic power series. We prove analogous statements for cycles of higher codimension, assuming a p-adic analogue of the variational Hodge conjecture, and prove that this analogue implies the usual variational Hodge conjecture. Applications are given to abelian schemes and to proper families of projective varieties.

Article information

Source
Duke Math. J., Volume 161, Number 11 (2012), 2167-2206.

Dates
First available in Project Euclid: 24 July 2012

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1343133926

Digital Object Identifier
doi:10.1215/00127094-1699490

Mathematical Reviews number (MathSciNet)
MR2957700

Zentralblatt MATH identifier
1248.14011

Subjects
Primary: 14C25: Algebraic cycles
Secondary: 14D07: Variation of Hodge structures [See also 32G20] 14F25: Classical real and complex (co)homology 14F30: $p$-adic cohomology, crystalline cohomology

Citation

Maulik, Davesh; Poonen, Bjorn. Néron–Severi groups under specialization. Duke Math. J. 161 (2012), no. 11, 2167--2206. doi:10.1215/00127094-1699490. https://projecteuclid.org/euclid.dmj/1343133926


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