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

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 On the degeneration of the Frölicher spectral sequence and small deformations

2020 ◽  
Vol 7 (1) ◽  
pp. 62-72
Author(s):  
Michele Maschio
Keyword(s):  
Spectral Sequence ◽  
Deformation Theory ◽  
Differential Operators ◽  
Complex Structure ◽  
Second Step ◽  
Complex Manifolds ◽  
Pseudo Differential Operators ◽  
Frölicher Spectral Sequence ◽  
Compact Complex Manifolds ◽  
Small Deformations

AbstractWe study the behavior of the degeneration at the second step of the Frölicher spectral sequence of a 𝒞∞ family of compact complex manifolds. Using techniques from deformation theory and adapting them to pseudo-differential operators we prove a result à la Kodaira-Spencer for the dimension of the second step of the Frölicher spectral sequence and we prove that, under a certain hypothesis, the degeneration at the second step is an open property under small deformations of the complex structure.

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