# 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.

## Key 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.

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Source: https://tomesphere.com/paper/1812.00865