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The study of harmonic differential forms on complex projective varieties, their invariantly defined filtrations, their integrals over topological cycles, especially over subvarieties, the deformations of these integrals and filtrations in families, and a multitude of generalizations.
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Are induced morphisms on cohomology strict with respect to the hodge filtration in the non K...
The following is a counterexample of what you ask for non-compact Kähler manifolds.
Let $X$ be a smooth projective variety and $D \subset X$ a very ample divisor. Since $U:= X- D$ is affine, the filt …