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Sets of special subvarieties of bounded degree

Part of: Cycles and subschemes

Published online by Cambridge University Press:  14 March 2023

David Urbanik
David Urbanik*
Affiliation:
Institut des Hautes Études Scientifiques, 35 Rte de Chartres, 91440 Bures-sur-Yvette, France [email protected]
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Abstract

Let $f : X \to S$ be a family of smooth projective algebraic varieties over a smooth connected quasi-projective base $S$, and let $\mathbb {V} = R^{2k} f_{*} \mathbb {Z}(k)$ be the integral variation of Hodge structure coming from degree $2k$ cohomology it induces. Associated to $\mathbb {V}$ one has the so-called Hodge locus $\textrm {HL}(S) \subset S$, which is a countable union of ‘special’ algebraic subvarieties of $S$ parametrizing those fibres of $\mathbb {V}$ possessing extra Hodge tensors (and so, conjecturally, those fibres of $f$ possessing extra algebraic cycles). The special subvarieties belong to a larger class of so-called weakly special subvarieties, which are subvarieties of $S$ maximal for their algebraic monodromy groups. For each positive integer $d$, we give an algorithm to compute the set of all weakly special subvarieties $Z \subset S$ of degree at most $d$ (with the degree taken relative to a choice of projective compactification $S \subset \overline {S}$ and very ample line bundle $\mathcal {L}$ on $\overline {S}$). As a corollary of our algorithm we prove conjectures of Daw–Ren and Daw–Javanpeykar–Kühne on the finiteness of sets of special and weakly special subvarieties of bounded degree.

Keywords

algebraic geometryHodge locidegrees

MSC classification

Primary: 14C30: Transcendental methods, Hodge theory , Hodge conjecture
Type
Research Article
Information
Compositio Mathematica , Volume 159 , Issue 3 , March 2023 , pp. 616 - 657
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Copyright
© 2023 The Author(s). The publishing rights in this article are licensed to Foundation Compositio Mathematica under an exclusive licence

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