Cohomological decomposition of compact complex manifolds and holomorphic deformations

Abstract

Abstract. The main goal of this note is the study of pureness and fullness properties of compact complex manifolds under holomorphic deformations. Firstly, we construct small deformations of pure-and-full complex manifolds along which one of these properties is lost while the other one is preserved. Secondly, we show that the property of being pure-and-full is not closed under holomorphic deformations. In order to do so, we focus on the class of 6-dimensional solvmanifolds endowed with invariant complex structures. In the special case of nilmanifolds, we also give a classification of those invariant complex structures that are both pure and full. In addition, relations of the cohomological decomposition with other metric and complex properties are studied.

Stability of Pseudo-Kähler Manifolds and Cohomological Decomposition

“…Note that all the complex structures above are nilpotent, so it follows from Proposition 4 (iii) that ∂ t ∂t γ t = 0, for every invariant 1-form γ t on X t . We next show, following again the ideas in [20], that X t is C ∞ -full for any t ∈ (−1, 1).…”

“…Due to the results in [20], the subspaces H ± (X t ) can be directly computed from the structure equations ( 9) (see Section 4 for more details). Indeed, one can show that X t is C ∞ -full only for t = 0.…”

“…Note that it suffices to prove the result for s ≥ 3. It is well-known (see for instance [20]) that for any such (g, J) there is a (1,0)-basis {ω 1 , ω 2 , ω 3 } satisfying equations of the form…”

On the stability of compact pseudo-Kähler and neutral Calabi-Yau manifolds

“…where the coefficients σ 12 , σ (X t ; R) be the subspace determined by the second de Rham cohomology classes that can be represented by closed real forms of bidegree (1, 1) on the compact complex manifold X t . As proved in [30,Proposition 3.4], this subspace is given by…”

On the stability of compact pseudo-Kähler and neutral Calabi-Yau manifolds

“…Let H + (X t ) ⊂ H 2 dR (X t ; R) be the subspace determined by the second de Rham cohomology classes that can be represented by closed real forms of bidegree (1, 1) on the compact complex manifold X t . As proved in [29,Proposition 3.4], this subspace is given by…”