Erratum to: The Frölicher spectral sequence can be arbitrarily non-degenerate

Bigalke, Laura; Rollenske, Sönke

doi:10.1007/s00208-013-0996-0

Cited by 13 publications

(28 citation statements)

References 2 publications

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“…Rearranging, we get Ω = ∂( Proof. Since d has a well-de ned bidegree we obtain a decomposition HA = p,q HA p,q , where HA p,q = (ker d ∩ A p,q )/ im d. Note that property (iv) implies that HA inherits properties (1) and (2). If ξ is the (n, n)-bidegree element used to de ne −, − on A, then we can de ne a bilinear pairing HA p,q ⊗ HA n−p,n−q −,− −→ C on the cohomology ring by [ .…”

Section: De Nition 23 (Cf [3]) a DI Erential On An N-dimensional Bmentioning

confidence: 99%

“…In the past few decades there has been interest in studying examples of compact complex manifolds for which the Frölicher spectral sequence degenerates only on the second page or later (see e.g. [2], [5]), with complex nilmanifolds being most amenable to this study. In this short note, we will observe through an immediate adaptation of the arguments in [3] that the symmetry dim E p,q = dim E n−p,n−q we have on the rst page of the spectral sequence, due to Serre duality on a compact complex n-manifold, continues to hold on all subsequent pages, thereby signi cantly reducing the amount of computation required to obtain these dimensions.…”

Section: Introductionmentioning

confidence: 99%

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Another proof of the persistence of Serre symmetry in the Frölicher spectral sequence

Milivojević

2020

Complex Manifolds

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Serre’s duality theorem implies a symmetry between the Hodge numbers, hp,q = hn−p,n−q, on a compact complex n–manifold. Equivalently, the first page of the associated Frölicher spectral sequence satisfies \dim E_1^{p,q} = \dim E_1^{n - p,n - q} for all p, q. Adapting an argument of Chern, Hirzebruch, and Serre [3] in an obvious way, in this short note we observe that this “Serre symmetry” \dim E_k^{p,q} = \dim E_k^{n - p,n - q} holds on all subsequent pages of the spectral sequence as well. The argument shows that an analogous statement holds for the Frölicher spectral sequence of an almost complex structure on a nilpotent real Lie group as considered by Cirici and Wilson in [4].

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Section: De Nition 23 (Cf [3]) a DI Erential On An N-dimensional Bmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Another proof of the persistence of Serre symmetry in the Frölicher spectral sequence

Milivojević

2020

Complex Manifolds

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“…In this Section we are going to discuss the existence of special Hermitian metrics and p-Kähler forms on the 2-step nilmanifolds with nilpotent complex structure constructed by Bigalke and Rollenske in [4]. In particular, for every n ≥ 2, these (4n−2)-dimensional compact complex manifolds are such that the Frölicher spectral sequence does not degenerate at the E n term.…”

Section: Special Hermitian Metrics On the Bigalke And Rollenske's Nil...mentioning

confidence: 99%

“…. , n) In [4] the authors show that that the Frölicher spectral sequence of X 4n−2 has nonvanishing differential d n , namely the Frölicher spectral sequence does not degenerate at the E n term.…”

Section: Special Hermitian Metrics On the Bigalke And Rollenske's Nil...mentioning

confidence: 99%

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$p$-Kähler and balanced structures on nilmanifolds with nilpotent complex structures

Sferruzza¹,

Tardini²

2021

Preprint

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Let (X, J) be a nilmanifold with a left-invariant nilpotent complex structure. We study the existence of p-Kähler structures (which include Kähler and balanced metrics) on X. More precisely, we determine an optimal p such that there are no p-Kähler structures on X. Finally, we show that, contrarily to the Kähler case, on compact complex manifolds there is no relation between the existence of balanced metrics and the degeneracy step of the Frölicher spectral sequence. More precisely, on balanced manifolds the degeneracy step can be arbitrarily large.

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