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Zigzags and the cohomology of complex manifolds

Banff International Research Station for Mathematical Innovation and Discovery

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.

Mathematics; Differential geometry; Algebraic geometry

video/mp4

Author affiliation: Ludwig Maximilian University

2020-04-27

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/74177

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/