On the structure of double complexes

Stelzig, Jonas (2021): On the structure of double complexes. In: Journal of the London Mathematical Society-Second Series, Vol. 104, No. 2: pp. 956-988

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 and 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 Poincare duality for higher pages of the Frolicher 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-Kahlerness degrees are not bimeromorphic invariants in dimensions higher than three and that the partial differential partial differential over bar -lemma and some related properties are bimeromorphic invariants if, and only if, they are stable under restriction to complex submanifolds.

• Item Type: Journal article
• Faculties: Mathematics, Computer Science and Statistics > Mathematics
• Subjects: 500 Science > 510 Mathematics
• ISSN: 0024-6107
• Language: English