Quasiprojectivity of images of mixed period maps

TL;DR

This paper proves that the closure of the image of any admissible mixed period map is quasiprojective, extending Griffiths' conjecture, using o-minimality and recent definability results for mixed period domains.

Contribution

It establishes the quasiprojectivity of mixed period map images and describes the structure of extension data parametrization, introducing new geometric and definability techniques.

Findings

The image of extension data for non-adjacent weights is quasi-affine.

Extensions of adjacent weights are parametrized by an ample theta bundle.

The proof employs o-minimality and definability in mixed period domains.

Abstract

We prove a mixed version of a conjecture of Griffiths: that the closure of the image of any admissible mixed period map is quasiprojective, with a natural ample bundle. Specifically, we consider the map from the image of the mixed period map to the image of the period map of the associated graded. On the one hand, we show in a precise manner that the parts of this map parametrizing extension data of non-adjacent-weight pure Hodge structures are quasi-affine. On the other hand, extensions of adjacent-weight pure polarized Hodge structures are parametrized by a compact complex torus (the intermediate Jacobian) equipped with a natural theta bundle which is ample in Griffiths transverse directions. Our proof makes heavy use of o-minimality, and recent work with B. Klingler associating a $\mathbb{R}_{an,exp}$-definable structure to mixed period domains and admissible mixed period maps.