The global derived period map

TL;DR

This paper extends the classical period map to a derived geometric setting, constructing a derived period domain and analyzing monodromy, with applications to derived Torelli problems and examples.

Contribution

It introduces the derived period domain and generalizes the classical period map within derived geometry, including monodromy analysis and derived Torelli problem insights.

Findings

Constructed the derived period domain classifying Hodge filtrations.

Established a canonical morphism from base to quotient of derived period domain by monodromy.

Discussed examples and the derived Torelli problem.

Abstract

Abstract. We develop the global period map in the context of derived geometry, generalising Griffiths' classical period map as well as the infinitesimal derived period map. We begin by constructing the derived period domain which classifies Hodge filtrations and enhances the classical period domain. We analyze the monodromy action. Then we associate to any polarized smooth projective map of derived stacks a canonical morphism of derived analytic stacks from the base into the quotient of the derived period domain by monodromy. We conclude the paper by discussing a few examples and a derived Torelli problem. In the appendix we describe how to present derived analytic Artin stacks as hypergroupoids, which may be of independent interest.