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- W3201469588 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." @default.
- W3201469588 created "2021-09-27" @default.
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- W3201469588 date "2023-03-01" @default.
- W3201469588 modified "2023-10-16" @default.
- W3201469588 title "Sets of special subvarieties of bounded degree" @default.
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- W3201469588 doi "https://doi.org/10.1112/s0010437x23007029" @default.
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