On cohomological decomposition of almost-complex manifolds and deformations

By Daniele Angella, Adriano Tomassini. 2009

48 citations recorded by the source corpus.

While small deformations of K\"ahler manifolds are K\"ahler too, we prove that the cohomological property to be $\mathcal{C}^\infty$-pure-and-full is not a stable condition under small deformations. This property, that has been recently introduced and studied by T.-J. Li and W. Zhang in [Comparing tamed and compatible symplectic cones and cohomological properties of almost complex manifolds, Comm. Anal. Geom.] and T. Dr\v{a}ghici, T.-J. Li, and W. Zhang in [Symplectic forms and cohomology decomposition of almost complex four-manifolds, Int. Math. Res. Not.], [On the J-anti-invariant cohomology of almost complex 4-manifolds, to appear in Q. J. Math.], is weaker than the K\"ahler one and characterizes the almost-complex structures that induce a decomposition in cohomology. We also study the stability of this property along curves of almost-complex structures constructed starting from the harmonic representatives in special cohomology classes.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/f4faca338453f893df466c5fdbf6ce35ad3c9033

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In Applied Mathematics

1. Daniele Angella
2. H. Kasuya
3. Weiyi Zhang
4. Tedi Drăghici
5. Tian-Jun Li
6. J. Stelzig

In Geometry and Topology

1. Daniele Angella
2. H. Kasuya
3. Weiyi Zhang
4. Tedi Drăghici
5. Tian-Jun Li
6. J. Stelzig
7. S. Bauer
8. Xiaokui Yang

Citations in the corpus

• On the J-anti-invariant cohomology of almost complex 4-manifolds (2011, background): “These subgroups are interesting almost complex invariants and there are several works already devoted to their study [12], [1], [11], [2].”
• Bott–Chern cohomology of solvmanifolds (2012, background): “(With the notation of [49, 9], this means that, in any case, Γ\G is complex-C∞-pure-and-full at every stage, namely, the de Rham cohomology admits a decomposition in pure-type subgroups with respect to the complex structure.”
• The Cohomologies of the Iwasawa Manifold and of Its Small Deformations (2012, background): “The Iwasawa manifold is one of the simplest example of non-Kähler manifold: indeed, being non-formal, see [12, page 158], it admits no Kähler structure; it has been studied by several authors as a fruitful source of interesting behaviors: see, for example, [12, 19, 1, 3, 29, 2].”