Inequalities à la Frölicher and cohomological decompositions
type: Item
instance of: scholarly article
title: Inequalities à la Frölicher and cohomological decompositions
zbMATH Open document ID: 1325.32018
DOI: 10.4171/JNCG/199
author: Daniele Angella
author: Adriano Tomassini
published in: Journal of Noncommutative Geometry
publication date: 2015-07-24T00:00:00Z
full work available at URL: https://arxiv.org/abs/1403.2298
review text: This paper is devoted to the Bott-Chern and Aeppli cohomologies of a vector space endowed with two anti-commuting endomorphisms squaring to zero. The main achievement is an inequality of Frölicher type relating the Bott-Chern, Aeppli and Dolbeault numbers, and its consequences. The applications considered here include the study of compact complex, symplectic, or generalized complex manifolds. Given a compact complex manifold \(X\), the Bott-Chern cohomology \(H_{BC}(X)\) and the Aeppli cohomology \(H_{A}(X)\) provide useful invariants, and have been studied in a number of contexts, see the paper's references. For instance if \(X\) is compact Kähler, or more generally if it satisfies the \(\partial\overline{\partial}\)-Lemma, \(H_{BC}, H_{A}\) are naturally isomorphic to the Dolbeault groups. \textit{D. Angella} and \textit{A. Tomassini} proved an important property of the BC cohomology of compact, complex (possibly non-Kähler) manifolds [Invent. Math. 192, No. 1, 71--81 (2013; Zbl 1271.32011)]; namely, that for any integer \(k\) \[ \sum_{p+q=k} \left( \dim_\mathbb{C} H^{p,q}_{BC}(X) + \dim_\mathbb{C} H^{p,q}_{BC}(X) \right) \;\geq\; 2\,\dim H^k_{dR}(X;\mathbb{C}), \] with equality if and only if \(X\) satisfies the \(\partial\overline{\partial}\)-Lemma. The present paper generalises that fact to a more algebraic framework, as follows. Let \(A\) be a \(\mathbb{Z}\)-graded \(\mathbb{K}\)-vector space endowed with endomorphisms \(\delta_1, \delta_2\) such that \(\delta_1^2 = \delta_2^2 = \delta_1\delta_2+\delta_2\delta_1 = 0\). Suppose both \(H_{(\delta_1 ; \delta_1)}\left(A\right)\) and \(H_{\left( \delta_2 ; \delta_2 \right)}\left(A\right)\) are finite-dimensional over \(\mathbb{K}\) in each degree. Then \[ \dim_\mathbb{K} H_{\left( \delta_1 , \delta_2 ; \delta_1\delta_2 \right)}\left(A\right) + \dim_\mathbb{K} H_{\left( \delta_1\delta_2 ; \delta_1 , \delta_2 \right)}\left(A\right) \;\geq\; \dim_\mathbb{K} H_{\left( \delta_1 ; \delta_1 \right)}\left(A\right) + \dim_\…
Mathematics Subject Classification ID: 32J18
Mathematics Subject Classification ID: 32J27
Mathematics Subject Classification ID: 53D05
Mathematics Subject Classification ID: 53D18
zbMATH DE Number: 6464168
zbMATH Keywords: compact complex manifolds
zbMATH Keywords: generalized complex structures
zbMATH Keywords: symplectic manifolds
zbMATH Keywords: \(\partial \overline{\partial}\)-lemma
zbMATH Keywords: Bott-Chern cohomology
zbMATH Keywords: Aeppli cohomology
reviewed by: Simon Chiossi
MaRDI profile type: Publication
arXiv ID: 1403.2298
cites work: Q5552286
cites work: Small Deformations of a Class of Compact Non-Kahler Manifolds
cites work: The cohomologies of the Iwasawa manifold and of its small deformations
cites work: On the \(\partial\overline{\partial}\)-lemma and Bott-Chern cohomology
cites work: Symplectical manifolds and cohomological decomposition
cites work: On cohomological decomposability of almost–Kähler structures
cites work: Q5601005
OpenAlex ID: W1502952944
description: scientific article