Leaf schemes and Hodge loci

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.

Four sections of this article have previously appeared in arxiv as independent articles. arXiv admin note: text overlap with arXiv:2408.11188, arXiv:2406.13432, arXiv:2403.11829, arXiv:2211.11405