The notion of cohomological mixed Hodge complex was introduced by Deligne [4] as a tool to construct the mixed Hodge structure on the cohomology of complex algebraic varieties. This is defined by using logarithmic forms and simplicial resolutions of varieties. It is called cohomological, because its conditions are imposed only after taking the global section functor. Forgetting the rational (or...

We prove a vanishing theorem for the Hodge number h 2;1 of projective toric varieties provided by a certain class of polytopes. We explain how this Hodge number also gives information about the deformation theory of the toric Gorenstein singularity derived from the same polytope. In particular, the vanishing theorem for h 2;1 implies that these deformations are unobstructed.

These notes are intended to accompany the course Introduction to Variations of Hodge Structure (VHS) at the 2010 ICTP Summer School on Hodge Theory. The modern theory of variations of Hodge structure (although some authors have referred to this period as the pre-history) begins with the work of Griffiths [23, 24, 25] and continues with that of Deligne [17, 18, 19], and Schmid [41]. The basic ob...