Sets of Special Subvarieties of Bounded Degree

TL;DR

This paper develops an algorithm to compute all weakly special subvarieties of a given degree in a family of algebraic varieties, proving finiteness conjectures related to special subvarieties.

Contribution

It introduces an explicit algorithm for enumerating weakly special subvarieties of bounded degree, advancing understanding of their structure and finiteness properties.

Findings

Algorithm for computing weakly special subvarieties of degree ≤ d

Proof of finiteness conjectures for special and weakly special subvarieties

Establishment of bounds on degrees of special subvarieties

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-K\"uhne on the finiteness of sets of special and weakly special subvarieties of bounded degree.