Néron–Severi groups under specialization

By Davesh Maulik, Bjorn Poonen. 2009

78 citations recorded by the source corpus.

Andr\'e used Hodge-theoretic methods to show that in a smooth proper family X to B of varieties over an algebraically closed field k of characteristic 0, 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 Andr\'e'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 nowhere p-adically 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.

Subfields: Geometry and Topology, Mathematical Physics, Pharmacology

View Paper: https://www.semanticscholar.org/paper/4afa919ff3d31c8da37b6651e05aeed8cd6565f9

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In Geometry and Topology

In Mathematical Physics

In Pharmacology

Citations in the corpus

Lectures on K3 Surfaces (2016, background): “See also the article [398] by Maulik and Poonen which contains a p-adic proof of this consequence.”
On the Picard number of K3 surfaces over number fields (2011, background): “A different approach to this problem can be found in [14].”