On occult period maps

TL;DR

This paper studies special period maps into ball quotients for moduli spaces of algebraic varieties, showing they are algebraically defined over their natural fields, connecting transcendental constructions to algebraic geometry.

Contribution

It interprets occult period maps as morphisms into moduli spaces of polarized abelian varieties and proves they are defined over the natural fields of the spaces.

Findings

Period maps are algebraically defined over their natural fields.

Connections established between transcendental constructions and algebraic geometry.

Provides a unified interpretation of occult period maps for various moduli spaces.

Abstract

We consider the "occult" period maps into ball quotients which exist for the moduli spaces of cubic surfaces, cubic threefolds, non-hyperelliptic curves of genus three and four. These were constructed in the work of Allcock/Carlson/Toledo, Looijenga/Swierstra, and Kondo. We interpret these maps as morphisms into moduli spaces of polarized abelian varieties of Picard type, and show that these morphisms, whose initial construction is transcendental, are defined over the natural field of definition of the spaces involved. This paper is extracted from section 15 of our paper arXiv:0912.3758, and differs from it only in some points of exposition.