  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.
