Hodge structures have been generalized for all complex varieties (even if they are singular and non-complete) in the form of mixed Hodge structures,...

In algebraic geometry, a mixed Hodge structure is an algebraic structure containing information about the cohomology of general algebraic varieties. It...

In mathematics, mixed Hodge modules are the culmination of Hodge theory, mixed Hodge structures, intersection cohomology, and the decomposition theorem...

of Hodge modules is a generalization. Roughly speaking, a mixed Hodge module on a variety X {\displaystyle X} is a sheaf of mixed Hodge structures over...

formula Borel–Moore homology Differential of the first kind Log structure Mixed Hodge structure Residue theorem Poincaré residue Deligne (1970), section II...

particular to an idea of mixed Hodge structure on singular varieties, and to deep analogies with étale cohomology. The Hodge conjecture on the 'middle'...

diamond Hodge duality Hodge filtration Hodge index theorem Hodge group Hodge star operator Hodge structure Mixed Hodge structure Hodge–Tate module Hodge theory...

the study of deformation theory. In addition, log structures serve to define the mixed Hodge structure on any smooth complex variety X, by taking a compactification...

(for varieties over C {\displaystyle \mathbb {C} } ) comes with a mixed Hodge structure, it is a differential-geometric invariant l-adic cohomology (over...

Mixed Hodge Structures, Ergebnisse der Mathematik und ihrer Grenzgebiete, Springer Verlag 2008 with Steven Zucker: Variation of mixed Hodge structure...