Families of Almost Complex Structures and Transverse (p, p)-Forms

Hind, Richard; Medori, Costantino; Томассини, Адриано

doi:10.1007/s12220-023-01391-x

Cited by 3 publications

(1 citation statement)

References 13 publications

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“…This theorem provides examples of (n + 1)-dimensional manifolds of Kodaira dimension 0 satisfying the ∂ ∂-Lemma but with no p-Kähler structures for p ≠ n. More in general, since N is a compact solvmanifold of completely solvable type different from a torus, by Hasegawa [10,Main Theorem] has no Kähler structure. For other results on p-Kähler manifolds see [18,13,8] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

On Kodaira dimension of almost complex 4-dimensional solvmanifolds without complex structures

2021

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The aim of this paper is to continue the study of Kodaira dimension for almost complex manifolds, focusing on the case of compact [Formula: see text]-dimensional solvmanifolds without any integrable almost complex structure. According to the classification theory we consider: [Formula: see text], [Formula: see text] and [Formula: see text] with [Formula: see text]. For the first solvmanifold we introduce special families of almost complex structures, compute the corresponding Kodaira dimension and show that it is no longer a deformation invariant. Moreover, we prove Ricci flatness of the canonical connection for the almost Kähler structure. Regarding the other two manifolds we compute the Kodaira dimension for certain almost complex structures. Finally, we construct a natural hypercomplex structure providing a twistorial description.

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Section: Introductionmentioning

confidence: 99%

On Kodaira dimension of almost complex 4-dimensional solvmanifolds without complex structures

2021

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$$\partial \bar{\partial }$$-lemma and p-Kähler structures on families of solvmanifolds

Cattaneo,

Tomassini

2024

Math. Z.

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We provide families of compact $$(n + 1)$$ ( n + 1 ) -dimensional complex non Kähler manifolds satisfying the $$\partial \bar{\partial }$$ ∂ ∂ ¯ -Lemma, with holomorphically trivial canonical bundle, carrying a balanced metric and with no p-Kähler structures. Such a construction extends to the completely solvable case in any dimension Nakamura’s construction of low-dimensional holomorphically parallelizable solvmanifolds.

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A note on p-Kähler structures on compact quotients of Lie groups

Fino,

Mainenti

2024

Annali di Matematica

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A p-Kähler structure on a complex manifold of complex dimension n is given by a d-closed transverse real (p, p)-form. In the paper, we study the existence of p-Kähler structures on compact quotients of simply connected Lie groups by discrete subgroups endowed with an invariant complex structure. In particular, we discuss the existence of p-Kähler structures on nilmanifolds, with a focus on the case $$p =2$$ p = 2 and complex dimension $$n = 4$$ n = 4 . Moreover, we prove that a $$(n-2)$$ ( n - 2 ) -Kähler almost abelian solvmanifold of complex dimension $$n\ge 3$$ n ≥ 3 has to be Kähler.

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Families of Almost Complex Structures and Transverse (p, p)-Forms

Cited by 3 publications

References 13 publications

On Kodaira dimension of almost complex 4-dimensional solvmanifolds without complex structures

On Kodaira dimension of almost complex 4-dimensional solvmanifolds without complex structures

$$\partial \bar{\partial }$$-lemma and p-Kähler structures on families of solvmanifolds

A note on p-Kähler structures on compact quotients of Lie groups

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