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This paper introduces a generalization of the ddc-condition for complexmanifolds. Like the dd^c-condition, it admits a diverse collection ofcharacterizations, and is hereditary under various geometric constructions.Most notably, it is an open property with respect to small deformations. Thecondition is satisfied by a wide range of complex manifolds including allcompact complex surfaces, and all compact Vaisman manifolds. We show there arecomputable invariants of a real homotopy type which in many cases prohibit itfrom containing any complex manifold satisfying such ddc-type conditions in lowdegrees. This gives rise to numerous examples of almost complex manifolds whichcannot be homotopy equivalent to any of these complex manifolds.

A ddc-type condition beyond the K\"ahler realm