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.
Dokumententyp: | Zeitschriftenartikel |
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Fakultät: | Mathematik, Informatik und Statistik > Mathematik |
Themengebiete: | 500 Naturwissenschaften und Mathematik > 510 Mathematik |
ISSN: | 0024-6107 |
Sprache: | Englisch |
Dokumenten ID: | 102327 |
Datum der Veröffentlichung auf Open Access LMU: | 05. Jun. 2023, 15:39 |
Letzte Änderungen: | 05. Jun. 2023, 15:39 |