Cohomological decomposition of compact complex manifolds and holomorphic deformations (Q2832833)

The authors study \textit{\(\mathcal{C}^\infty\)-pure-and-full} complex manifolds in the sense of [\textit{T.-J. Li} and \textit{W. Zhang}, Commun. Anal. Geom. 17, No. 4, 651--683 (2009; Zbl 1225.53066)], namely, complex manifolds for which the second de Rham cohomology group decomposes into the direct sum of the subgroups represented by invariant, respectively anti-invariant forms with respect to the complex structure. The properties of decomposability into a (possibly, non-direct) sum, respectively of trivial intersections between the above subgroups are called \textit{\(\mathcal{C}^\infty\)-full}, respectively \textit{\(\mathcal{C}^\infty\)-pure}. The interest in such a decomposition is related to the comparison between the tamed and the compatible cones and to a question by Donaldson for almost-complex \(4\)-manifolds.NEWLINENEWLINEIn particular, the authors study the behaviour of the \(\mathcal{C}^\infty\)-pure-and-full property under holomorphic deformations. They provide examples to show that the \(\mathcal{C}^\infty\)-pure and \(\mathcal{C}^\infty\)-full properties are not closed under holomorphic deformations, see Corollary 4.2. As a difference with the case of dimension \(4\), they give examples where the dimension of the invariant subgroup with respect to the complex structure does not have the lower-semi-continuity property, see Proposition 3.1. They show that the \(\mathcal{C}^\infty\)-pure-and-full property is unrelated with the existence of special metrics as SKT (pluriclosed), locally conformal Kähler, balanced, strongly Gauduchon, see Corollary 5.2. The examples are among \(6\)-dimensional solvmanifolds. In particular, \(\mathcal{C}^\infty\)-pure-and-full \(6\)-dimensional nilmanifolds are classified in Section 2. The authors observe several connections, e.g., on \(6\)-dimensional nilmanifolds between \(\mathcal{C}^\infty\)-pure-and-fullness and the maximal values of the \(\Delta\)-invariant measuring the difference between the dimension of the Bott-Chern cohomology and Betti numbers.