scholarly journals Invariant Complex Structures on 6-Nilmanifolds: Classification, Frölicher Spectral Sequence and Special Hermitian Metrics

2014 ◽  
Vol 26 (1) ◽  
pp. 252-286 ◽  
Author(s):  
M. Ceballos ◽  
A. Otal ◽  
L. Ugarte ◽  
R. Villacampa
Keyword(s):  
Spectral Sequence ◽  
Complex Structures ◽  
Hermitian Metrics ◽  
Frölicher Spectral Sequence

scholarly journals Higher-page Bott–Chern and Aeppli cohomologies and applications

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Dan Popovici ◽  
Jonas Stelzig ◽  
Luis Ugarte
Keyword(s):  
Positive Integer ◽  
Spectral Sequence ◽  
Complex Manifolds ◽  
Serre Duality ◽  
Frölicher Spectral Sequence ◽  
Compact Complex Manifolds

Abstract For every positive integer r, we introduce two new cohomologies, that we call E r {E_{r}} -Bott–Chern and E r {E_{r}} -Aeppli, on compact complex manifolds. When r = 1 {r\kern-1.0pt=\kern-1.0pt1} , they coincide with the usual Bott–Chern and Aeppli cohomologies, but they are coarser, respectively finer, than these when r ≥ 2 {r\geq 2} . They provide analogues in the Bott–Chern–Aeppli context of the E r {E_{r}} -cohomologies featuring in the Frölicher spectral sequence of the manifold. We apply these new cohomologies in several ways to characterise the notion of page- ( r - 1 ) {(r-1)} - ∂ ⁡ ∂ ¯ {\partial\bar{\partial}} -manifolds that we introduced very recently. We also prove analogues of the Serre duality for these higher-page Bott–Chern and Aeppli cohomologies and for the spaces featuring in the Frölicher spectral sequence. We obtain a further group of applications of our cohomologies to the study of Hermitian-symplectic and strongly Gauduchon metrics for which we show that they provide the natural cohomological framework.

scholarly journals Degeneration at E2 of certain spectral sequences

2016 ◽  
Vol 27 (14) ◽  
pp. 1650111 ◽  
Author(s):  
Dan Popovici
Keyword(s):  
Spectral Sequence ◽  
Differential Operators ◽  
Spectral Gap ◽  
Sufficient Conditions ◽  
Main Idea ◽  
Second Step ◽  
Compact Complex Manifold ◽  
Frölicher Spectral Sequence ◽  
Foliated Structure ◽  
Laplace Type

We propose a Hodge theory for the spaces [Formula: see text] featuring at the second step either in the Frölicher spectral sequence of an arbitrary compact complex manifold [Formula: see text] or in the spectral sequence associated with a pair [Formula: see text] of complementary regular holomorphic foliations on such a manifold. The main idea is to introduce a Laplace-type operator associated with a given Hermitian metric on [Formula: see text] whose kernel in every bidegree [Formula: see text] is isomorphic to [Formula: see text] in either of the two situations discussed. The surprising aspect is that this operator is not a differential operator since it involves a harmonic projection, although it depends on certain differential operators. We then use this Hodge isomorphism for [Formula: see text] to give sufficient conditions for the degeneration at [Formula: see text] of the spectral sequence considered in each of the two cases in terms of the existence of certain metrics on [Formula: see text]. For example, in the Frölicher case, we prove degeneration at [Formula: see text] if there exists an SKT metric [Formula: see text] (i.e. such that [Formula: see text]) whose torsion is small compared to the spectral gap of the elliptic operator [Formula: see text] defined by [Formula: see text]. In the foliated case, we obtain degeneration at [Formula: see text] under a hypothesis involving the Laplacians [Formula: see text] and [Formula: see text] associated with the splitting [Formula: see text] induced by the foliated structure.

scholarly journals A general description of the terms in the Frölicher spectral sequence

1997 ◽  
Vol 7 (1) ◽  
pp. 75-84 ◽  
Author(s):  
Luis A Cordero ◽  
Marisa Fernández ◽  
Luis Ugarte ◽  
Alfred Gray
Keyword(s):  
Spectral Sequence ◽  
General Description ◽  
Frölicher Spectral Sequence

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

2020 ◽  
Vol 7 (1) ◽  
pp. 141-144
Author(s):  
Aleksandar Milivojević
Keyword(s):  
Spectral Sequence ◽  
Lie Group ◽  
Duality Theorem ◽  
Complex Structure ◽  
Short Note ◽  
Almost Complex Structure ◽  
Analogous Statement ◽  
Almost Complex ◽  
Hodge Numbers ◽  
Frölicher Spectral Sequence

AbstractSerre’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].

scholarly journals The Frölicher spectral sequence for compact nilmanifolds

1991 ◽  
Vol 35 (1) ◽  
pp. 56-67 ◽  
Author(s):  
Luis A. Cordero ◽  
Marisa Fernández ◽  
Alfred Gray
Keyword(s):  
Spectral Sequence ◽  
Frölicher Spectral Sequence ◽  
Compact Nilmanifolds

scholarly journals Holomorphic Poisson Cohomology

2015 ◽  
Vol 2 (1) ◽  
Author(s):  
Zhuo Chen ◽  
Daniele Grandini ◽  
Yat-Sun Poon
Keyword(s):  
Spectral Sequence ◽  
Complex Structure ◽  
Complex Structures ◽  
Dolbeault Cohomology ◽  
Double Complex ◽  
Poisson Cohomology ◽  
Compact Complex Surfaces ◽  
Generalized Geometry ◽  
Holomorphic Poisson Cohomology ◽  
Compact Complex Manifolds

AbstractHolomorphic Poisson structures arise naturally in the realm of generalized geometry. A holomorphic Poisson structure induces a deformation of the complex structure in a generalized sense, whose cohomology is obtained by twisting the Dolbeault @-operator by the holomorphic Poisson bivector field. Therefore, the cohomology space naturally appears as the limit of a spectral sequence of a double complex. The first sheet of this spectral sequence is simply the Dolbeault cohomology with coefficients in the exterior algebra of the holomorphic tangent bundle. We identify various necessary conditions on compact complex manifolds on which this spectral sequence degenerates on the level of the second sheet. The manifolds to our concern include all compact complex surfaces, Kähler manifolds, and nilmanifolds with abelian complex structures or parallelizable complex structures.

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

2014 ◽  
Vol 358 (3-4) ◽  
pp. 1119-1123 ◽  
Author(s):  
Laura Bigalke ◽  
Sönke Rollenske
Keyword(s):  
Spectral Sequence ◽  
Frölicher Spectral Sequence
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