Neron–Severi groups under specialization
Name
Poonen_Neron-severi groups.pdf
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480.01 KB
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Checksum (MD5)
2a57cd81895c8c100c313601a0bb4645
Author(s)
Maulik, Davesh
Poonen, Bjorn
Date Issued
August 2012
Journal
Duke Mathematical Journal
Publisher
Duke University Press
Citation
Maulik, Davesh, and Bjorn Poonen. “Néron–Severi groups under specialization.” Duke Mathematical Journal 161, no. 11 (August 2012): 2167-2206.
Version
Author's final manuscript
Abstract
Andre used Hodge-theoretic methods to show that in a smooth proper family X → B 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.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
Terms of Use
Creative Commons Attribution-Noncommercial-Share Alike 3.0
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DOI of Published Version
https://doi.org/10.1215/00127094-1699490
