DGA-Models of variations of mixed Hodge structures

We define objects over Morgan's mixed Hodge diagrams which will be algebraic models of unipotent variations of mixed hodge structures over K\"ahler manifolds. As an analogue of Hain-Zucker's equivalence between unipotent variations of mixed Hodge structures and mixed Hodge representations of the fundamental group with Hain's mixed hodge structure, we give an equivalence between the category of our VMHS-like objects and the category of mixed Hodge representations of the dual Lie algebra of Sullivan's minimal model with Morgan's mixed Hodge structure. By this result, we can put various (tannakian theoretical) non-abelian mixed Hodge structures on the category of our new objects like the taking fibers of variations of mixed Hodge structures at points. By certain modifications of the result, we also give models of non-unipotent variations of mixed Hodge structures.Comment: 30 page