A period map for global derived stacks

TL;DR

This paper extends the Griffiths period map to derived algebraic geometry, constructing a global version of the local period map as a morphism of derived stacks, linking smooth projective varieties to Hodge structures.

Contribution

It develops a derived algebraic geometric framework for the period map, completing the description of local maps and lifting them to global derived stacks.

Findings

Constructed a global derived stack version of the period map.

Connected local deformation functors to global derived stacks.

Enhanced understanding of Hodge structures in derived geometry.

Abstract

We develop the theory of Griffiths period map, which relates the classification of smooth projective varieties to the associated Hodge structures, in the framework of Derived Algebraic Geometry. We complete the description of the local period map as a morphism of derived deformation functors, following the path marked by Fiorenza, Manetti and Martinengo. In the end we show how to lift the local period map to a (non-geometric) morphism of derived stacks, in order to construct a global version of that.