Abstract
Abstract This paper introduces a generalization of the $dd^c$ -condition for complex manifolds. Like the $dd^c$ -condition, it admits a diverse collection of characterizations, and is hereditary under various geometric constructions. Most notably, it 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. We show there are computable invariants of a real homotopy type which in many cases prohibit it from containing any complex manifold satisfying such $dd^c$ -type conditions in low degrees. This gives rise to numerous examples of almost complex manifolds which cannot be homotopy equivalent to any of these complex manifolds.
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