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BIRS Workshop Lecture Videos

Zigzags and the cohomology of complex manifolds Stelzig, Jonas


Deligne, Griffiths, Morgan and Sullivan famously characterised the $\partial\bar\partial$-Lemma as by the following property: The double complex of forms decomposes as a direct sum of two kinds of irreducible subcomplexes: 'Squares' and 'dots', where only the latter contribute to cohomology. In this talk, we explore the implications of the following folklore generalisation of this: Every (suitably bounded) double complex decomposes into irreducible complexes and these are 'squares' and 'zigzags', with a dot being a zigzag of length 1. This yields insight into the structure of and relation between the various cohomology groups. Applied to complex manifolds, we obtain, among others, Serre duality for all pages of the FSS, a three space decomposition on the middle cohomology and new bimeromorphic invariants. We end the talk with several open questions.

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