Deformation Openness and Closedness of Various Classes of Compact Complex Manifolds; Examples

By Dan Popovici. 2011

54 citations recorded by the source corpus.

We review the relations between compact complex manifolds carrying various types of Hermitian metrics (K\"ahler, balanced or {\it strongly Gauduchon}) and those satisfying the $\partial\bar\partial$-lemma or the degeneration at $E_1$ of the Fr\"olicher spectral sequence, as well as the behaviour of these properties under holomorphic deformations. The emphasis will be placed on the notion of {\it strongly Gauduchon} (sG) manifolds that we introduced recently in the study of deformation limits of projective and Moishezon manifolds. Various examples of sG and non-sG manifolds are exhibited while a range of constructions already known in the literature are reviewed and reinterpreted from this new standpoint.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/cad701ccc3ecfeb2433d7a2551f24e00c864ba3f

How researchers see this paper

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In Applied Mathematics

1. Anna Fino
2. Xiaokui Yang
3. Daniele Angella
4. L. Ugarte
5. Kefeng Liu
6. A. Otal
7. H. Kasuya
8. Valentino Tosatti

In Geometry and Topology

1. Anna Fino
2. Xiaokui Yang
3. Daniele Angella
4. L. Ugarte
5. Kefeng Liu
6. A. Otal
7. H. Kasuya
8. Valentino Tosatti

Citations in the corpus

• Ricci Curvatures on Hermitian manifolds (2014, background): “For more details, we refer to [2, 9, 45, 46] and also the references therein.”
• Bott–Chern cohomology of solvmanifolds (2012, background): “, [56]: namely, a property P is said to be (Zariski-)closed if, for any family {Xt}t∈∆ of compact complex manifolds such that P holds for any t ∈ ∆ \ {0} in the punctured-disk, then P holds also for X0.”
• Six-Dimensional Solvmanifolds with Holomorphically Trivial Canonical Bundle (2014, background): “More concretely, there exists a holomorphic family of compact complex manifolds (M,Ja)a∈∆ such that (M,Ja) has balanced metric for any a 6= 0 but the central limit (M,J0) does not admit any sG metric, which provides a counterexample to the Popovici and Demailly closedness conjectures formulated in [”