On the Structure of Double Complexes

Abstract

We study consequences and applications of the folklore statement that every double complex over a field decomposes into so-called squares and zigzags. This result makes questions about the associated cohomology groups and spectral sequences easy to understand. We describe a notion of `universal' quasi-isomorphism, investigate the behaviour of the decomposition under tensor product and compute the Grothendieck ring of the category of bounded double complexes over a field with finite cohomologies up to such quasi-isomorphism (and some variants). Applying the theory to the double complexes of smooth complex valued forms on compact complex manifolds, we obtain a Poincaré duality for higher pages of the Frölicher spectral sequence, construct a functorial three-space decomposition of the middle cohomology, give an example of a map between compact complex manifolds which does not respect the Hodge filtration strictly, compute the Bott-Chern and Aeppli cohomology for Calabi-Eckmann manifolds, introduce new numerical bimeromorphic invariants, show that the non-Kählerness degrees are not bimeromorphic invariants in dimensions higher than three and that the $\partial \overline{\partial }$-lemma and some related properties are bimeromorphic invariants if, and only if, they are stable under restriction to complex submanifolds.

arXiv commentfinal version; to appear in J. London Math. Soc

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Cited by in corpus (6)

• Deformations of Dolbeault cohomology classes
• Dolbeault and Bott-Chern formalities: deformations and $\partial \overline{\partial }$-lemma
• Pluripotential Homotopy Theory
• The $\partial \bar{\partial }$-lemma under surjective maps
• Frölicher spectral sequence of compact complex manifolds with special Hermitian metrics
• Some computations on trivial canonical-bundle solvmanifolds