A \(dd^c\)-type condition beyond the Kähler realm (Q6665326)

The usual \(\partial \overline{\partial}\)-condition is a versatile tool in the study of compact complex manifolds for at least the following reasons:\N\begin{itemize}\N\item[(1)] It admits characterizations of rather distinct nature (using elements, indecomposable bicomplexes, the Frölicher spectral sequence and pure Hodge structures, numerical inequalities).\N\item[(2)] It passes to other manifolds in many geometric situations, such as holomorphic domination, projective bundles, small deformations, blow-ups (along \(\partial \overline{\partial}\)-centres), etc.\N\item[(3)] It holds on a fairly large class of manifolds, in particular, on compact Kähler manifolds.\N\item[(4)] It implies topological restrictions on the underlying manifold: odd Betti numbers are even, and formality holds, in the sense of rational homotopy theory.\N\end{itemize}\NIn the paper under review, the authors present a generalization of the \(\partial \overline{\partial}\)-condition, termed the \(\partial \overline{\partial}+3\)-condition, for which they obtain full analogues of (1)--(3) above.\N\NTo make (3) in the \(\partial \overline{\partial}+3\)-condition, the authors introduce a nonnegative integer that measures the extent to which the pure Hodge condition fails, called the purity defect (in Definition 3.15). Most notably, the \(\partial \overline{\partial}+3\)-condition is an open property with respect to small deformations. The condition is satisfied by a wide range of complex manifolds, including all compact complex surfaces, and all compact Vaisman manifolds. In the last section, the authors study topological obstructions (with various examples) to the existence of complex structures satisfying a low-degree variant of the \(\partial \overline{\partial}+3\)-condition.