The geometry of the Deligne-Hodge decomposition
In this thesis we explore variations of mixed Hodge structures, both locally and asymptotically. First, making use of certain distinguished gradings of the Hodge and weight filtrations, I compute the curvature of appropriate classifying spaces and Hodge bundles relative to a natural “mixed” Hodge metric. Second, I obtain appropriate generalizations of the Nilpotent Orbit Theorem for admissible variations, norm estimates, a version of the invariant cycle Theorem, and extend an equivalence of categories theorem of Deligne. Third, I demonstrate the existence of canonical Higgs fields associated to such variations and discuss their relations with a partial interpretation of Mirror Symmetry due to Deligne.