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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.
9
votes
How do non-trivial global differentials give non-trivial cohomology classes in positive char...
Although Felipe answered your question, let me come back to David's wish, and give an answer using and advocating crystalline cohomology:
First, if $X$ is, say, smooth and proper over a field $k$, t …