15 August 2012 Néron–Severi groups under specialization
Davesh Maulik, Bjorn Poonen
Duke Math. J. 161(11): 2167-2206 (15 August 2012). DOI: 10.1215/00127094-1699490


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.


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Davesh Maulik. Bjorn Poonen. "Néron–Severi groups under specialization." Duke Math. J. 161 (11) 2167 - 2206, 15 August 2012. https://doi.org/10.1215/00127094-1699490


Published: 15 August 2012
First available in Project Euclid: 24 July 2012

zbMATH: 1248.14011
MathSciNet: MR2957700
Digital Object Identifier: 10.1215/00127094-1699490

Primary: 14C25
Secondary: 14D07 , 14F25 , 14F30

Rights: Copyright © 2012 Duke University Press


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Vol.161 • No. 11 • 15 August 2012
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