On the Structure of Double Complexes

TL;DR

This paper explores the structure of double complexes over fields, demonstrating their decomposition into squares and zigzags, and applies this to complex geometry to derive new invariants and dualities.

Contribution

It introduces the notion of universal quasi-isomorphism, analyzes decomposition behavior under tensor products, and computes the Grothendieck ring for bounded double complexes, with applications to complex manifolds.

Findings

Decomposition of double complexes into squares and zigzags simplifies cohomology analysis.

Established Poincaré duality for higher pages of the Frölicher spectral sequence.

Computed Bott-Chern and Aeppli cohomology for Calabi-Eckmann manifolds.

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 Poincar\'e duality for higher pages of the Fr\"olicher 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-K\"ahlerness 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.