Families of almost complex structures and transverse $(p,p)$-forms

Abstract

An almost p-Kähler manifold is a triple (M, J, Ω), where (M, J) is an almost complex manifold of real dimension 2n and Ω is a closed real tranverse (p, p)-form on (M, J), where 1 ≤ p ≤ n. When J is integrable, almost p-Kähler manifolds are called p-Kähler manifolds. We produce families of almost p-Kähler structures (Jt, Ωt) on C 3 , C 4 , and on the real torus T 6 , arising as deformations of Kähler structures (J0, g0, ω0), such that the almost complex structures Jt cannot be locally compatible with any symplectic form for t = 0. Furthermore, examples of special compact nilmanifolds with and without almost p-Kähler structures are presented. Contents 1. Introduction 1 2. Almost p-Kähler structures 4 3. Curves of almost complex structures preserving the 2 th -power 6 4. Semi-Kähler deformations of balanced metrics 10 5. Applications and examples 13 References 19

p-Kähler and balanced structures on nilmanifolds with nilpotent complex structures

“…Recently, their behaviour under small deformations of the complex structure has been studied in [19]. In [13] such forms have been extended to nonintegrable almost-complex manifolds and we recall here the following lemma that provides an obstruction to their existence, see [13,Proposition 3.4].…”

“…Lemma 2.2 [13] Let (X , J ) be a compact complex manifold of complex dimension n. Suppose that there exists a non-closed (2n…”

$p$-Kähler and balanced structures on nilmanifolds with nilpotent complex structures

“…Recently, their behaviour under small deformations of the complex structure has been studied in [17]. In [11] such forms have been extended to nonintegrable almost-complex manifolds and we recall here the following lemma that provides an obstruction to their existence, see [11,Proposition 3.4].…”