On Cohomological Decomposition of Almost-Complex Manifolds and Deformations

TL;DR

This paper investigates the stability of a specific cohomological property of almost-complex manifolds under small deformations, revealing that unlike Kähler stability, this property is not preserved, and explores conditions for its stability.

Contribution

It introduces and analyzes the stability of the ^ ext{-}full property in almost-complex manifolds, showing it is not stable under small deformations and studying conditions for stability.

Findings

The ^ ext{-}full property is not stable under small deformations.

Ke4hler manifolds maintain their properties under small deformations.

Conditions for stability of the cohomological decomposition are identified.

Abstract

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.