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 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 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 ∂∂lemma and some related properties are bimeromorphic invariants if, and only if, they are stable under restriction to complex submanifolds.

On metric and cohomological properties of Oeljeklaus-Toma manifolds

“…In particular, Betti numbers, Hodge numbers, Bott-Chern and Aeppli numbers are linear combinations of the multiplicities mult Z (A) for zigzags Z, see [S21, Corollary B]. We also recall that an E 1 -isomorphism is a morphism of bounded double complexes inducing an isomorphism on the first page of the Frölicher spectral sequence, see [S21,Definition D]. By [S21, Proposition E], the E 1 -isomorphism type of a bounded double complex is completely described by the multiplicities of all non-projective indecomposable bicomplexes (i.e.…”

Bigraded notions of formality and Aeppli-Bott-Chern-Massey products

“…bigraded complex vector spaces with differentials, suggestively denoted ∂ and ∂, of bidegree (1, 0) and (0, 1) respectively, such that (∂ + ∂) 2 = 0. We recall that (bounded) double complexes admit direct sum decompositions into well-understood indecomposable subcomplexes ("squares and zigzags"), see [KQ20], [Ste21].…”

Deformations of higher-page analogues of $$\partial {{\bar{\partial }}}$$-manifolds

“…We may therefore start the proof of Theorem 1.2 by picking η 1 ∈ C. By property (2), ψ 1 ∈ T . Again by property (1), the inclusion C → A X induces isomorphisms in Bott-Chern cohomology and higher pages of the Frölicher spectral sequence of C and A X [17,Cor. 13], whenever a form in C is exact in any way (w.r.t.…”

“…Recall[17] that this means it induces an isomorphism both in Dolbeault and conjugate Dolbeault cohomology (the latter being automatic if C is a real sub-complex).…”