Leaf schemes and Hodge loci

TL;DR

This collection explores the arithmetic properties of differential equations, foliations, and Hodge loci, proposing a new theory of leaf schemes with a local-global conjecture, connecting algebraic and geometric structures.

Contribution

It introduces the concept of leaf schemes over finitely generated rings and formulates a related local-global conjecture, advancing the understanding of arithmetic and geometric properties of foliations.

Findings

Development of a theory of leaf schemes over finitely generated rings

Formulation of a local-global conjecture for leaf schemes

Insights into the arithmetic properties of Hodge loci and differential equations

Abstract

This is a collection of articles, written as sections, on arithmetic properties of differential equations, holomorphic foliations, Gauss-Manin connections and Hodge loci. Each section is independent from the others and it has its own abstract and introduction and the reader might get an insight to the text by reading the introduction of each section. The main connection between them is through comments in footnotes. Our major aim is to develop a theory of leaf schemes over finitely generated subrings of complex numbers, such that the leaves are also equipped with a scheme structure. We also aim to formulate a local-global conjecture for leaf schemes.