Inequalities à la Frölicher and cohomological decompositions
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 \[ \dim_\mathbb{K} H_{\left( \delta_1 , \delta_2 ; \delta_1\delta_2 \right)}\left(A\right) + \dim_\mathbb{K} H_{\left( \delta_1\delta_2 ; \delta_1 , \delta_2 \right)}\left(A\right) \;\geq\; \dim_\mathbb{K} H_{\left( \delta_1 ; \delta_1 \right)}\left(A\right) + \dim_\mathbb{K} H_{\left( \delta_2 ; \delta_2 \right)}\left(A\right) \] in each degree. Furthermore, the above is an equality whenever the \(\delta_1\delta_2\)-Lemma is true. The first application is to compact symplectic manifolds \((X, \omega)\). Defining \(\partial^\Lambda := \left[\partial,\, -\iota_{\omega^{-1}}\right]\), the authors prove that \[ \dim_\mathbb{R} H^{\bullet}_{\left( \partial, \partial^\Lambda ; \partial\partial^\Lambda \right)}\left(X\right) + \dim_\mathbb{R} H^{\bullet}_{\left( \partial\partial^\Lambda ; \partial, \partial^\Lambda \right)}\left(X\right) \;\geq\; 2\, \dim H^{\bullet}_{dR}(X;\mathbb{R}), \] and equality holds iff \(X\) satisfies the hard Lefschetz condition. Secondly, let \(X\) be a smooth compact manifold endowed with an \(H\)-twisted generalized complex structure \(\mathcal{J}\). Then the relevant cohomology theories \(GH\) fulfill \[ \dim_\mathbb{C} GH^{\bullet}_{BC_{\mathcal{J},H}}(X) + \dim_\mathbb{C} GH^{\bullet}_{A_{\mathcal{J},H}}(X) \;\geq\; \dim_\mathbb{C} GH^{\bullet}_{\overline{\partial}_{\mathcal{J},H}}(X) + \dim_\mathbb{C} GH^{\bullet}_{\partial_{\mathcal{J},H}}(X). \] In addition, \(X\) satisfies the \((\partial{\mathcal{J},H}\overline{\partial}_{\mathcal{J},H})\)-Lemma if and only if the Hodge and Frölicher spectral sequences associated to a suitable canonical complex degenerate at the first level and equality holds.
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