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$dd^c$-TYPE CONDITION BEYOND THE KÄHLER REALM
- Part of
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- Journal:
- Journal of the Institute of Mathematics of Jussieu / Volume 23 / Issue 4 / July 2024
- Published online by Cambridge University Press:
- 28 November 2023, pp. 1651-1704
- Print publication:
- July 2024
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- Article
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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.
