Duke Mathematical Journal
- Duke Math. J.
- Volume 161, Number 11 (2012), 2167-2206.
Néron–Severi groups under specialization
André used Hodge-theoretic methods to show that in a smooth proper family of varieties over an algebraically closed field of characteristic zero, there exists a closed fiber having the same Picard number as the geometric generic fiber, even if 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 , the locus on the base where the Picard number jumps is -adically nowhere dense. Our proof uses the “-adic Lefschetz -theorem” of Berthelot and Ogus, combined with an analysis of -adic power series. We prove analogous statements for cycles of higher codimension, assuming a -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.
Duke Math. J., Volume 161, Number 11 (2012), 2167-2206.
First available in Project Euclid: 24 July 2012
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
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
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