Deformations and rigidity for mixed period maps

In this article, we prove a rigidity criterion for period maps of admissible variations of graded-polarizable mixed Hodge structure, and establish rigidity in a number of cases, including families of quasi-projective curves, projective curves with ordinary double points, the complement of the canonical curve in families of Kynev--Todorov surfaces, period maps attached to the fundamental groups of smooth varieties and normal functions.