Local-global principle for leaf schemes

TL;DR

This paper explores the structure of Hodge loci as leaf schemes within foliations, utilizing the Gauss-Manin connection, and investigates a conjecture linking local-global principles to the algebraicity of leaf schemes.

Contribution

It introduces a local-global principle for leaf schemes and examines its implications for the algebraicity of Hodge loci, extending existing theorems.

Findings

Establishes a conjecture connecting local-global principles to leaf scheme algebraicity.

Analyzes the Gauss-Manin connection matrix in the context of Hodge loci.

Proposes a framework for understanding leaf schemes through foliations.

Abstract

We study Hodge loci as leaf schemes of foliations. The main ingredient is the Gauss-Manin connection matrix of families of projective varieties. We also aim to investigate a conjecture on the ring of definition of leaf schemes and its consequences such as the algebraicity of leaf schemes (Cattani-Deligne-Kaplan theorem in the case of Hodge loci). This conjecture is a consequence of a local-global principle for leaf schemes.