Quasi-modular forms attached to Hodge structures

TL;DR

This paper introduces quasi-automorphic forms related to Hodge structures, extending the concept of quasi-modular forms by examining the moduli space of polarized lattices and their automorphism groups.

Contribution

It generalizes quasi-modular forms to Hodge structures through a new perspective on moduli spaces and automorphism group actions.

Findings

Defined quasi-automorphic forms in the context of Hodge structures

Connected the theory to algebraic groups acting on polarized lattices

Extended classical quasi-modular forms to a broader geometric setting

Abstract

The space $D$ of Hodge structures on a fixed polarized lattice is known as Griffiths period domain and its quotient by the isometry group of the lattice is the moduli of polarized Hodge structures of a fixed type. When $D$ is a Hermition symmetric domain then we have automorphic forms on $D$, which according to Baily-Borel theorem, they give an algebraic structure to the mentioned moduli space. In this article we slightly modify this picture by considering the space $U$ of polarized lattices in a fixed complex vector space with a fixed Hodge filtration and polarization. It turns out that the isometry group of the filtration and polarization, which is an algebraic group, acts on $U$ and the quotient is again the moduli of polarized Hodge structures. This formulation leads us to the notion of quasi-automorphic forms which generalizes quasi-modular forms attached to elliptic curves.