Deformation stability of p-SKT and p-HS manifolds

Abstract

In this paper, we introduce the notions of p-Hermitian-symplectic and p-pluriclosed compact complex manifolds as generalisations for an arbitrary positive integer p not exceeding the complex dimension of the manifold of the standard notions of Hermitian-symplectic and SKT manifolds that correspond to the case p = 1. We then notice that these two properties are equivalent on ∂∂-manifolds and go on to prove that in (smooth) complex analytic families of ∂∂-manifolds, they are deformation open. Concerning closedness results, we prove that the cones A p , resp. C p , of Aeppli cohomology classes of strictly weakly positive (p, p)-forms Ω that are p-pluriclosed, resp. p-Hermitian-symplectic, must be equal on the limit fibre if they are equal on the other fibres and if some rather weak ∂∂-type assumptions are made on the other fibres.

Co-polarised Deformations of Gauduchon Calabi-Yau $\partial\bar{\partial}$-Manifolds and Deformation of $p$-SKT $h$-$\partial\bar{\partial}$-Manifolds

“…where v 0 is a unique primitive C ∞ (1, 2)-form and ζ is a unique C ∞ vector field of type (1,0). Applying ∂ to α, one obtains (27).…”

“…By Wu's result [15], the ∂ ∂property is deformation open. On the other hand, It is proved, in ( [1], Conclusion 4.4), that the notion of p-pluriclosed (or briefly p-SKT if there exists a strictly weakly positive (p, p)-form for p ∈ {0, • • • , n} that is ∂ ∂-closed) compact complex ∂ ∂-manifold is open under small deformations. In the case where p = n − 1, the p-SKT property is nothing but Gauduchon metric, which means that the Gauduchon ∂ ∂-manifold is also open under holomorphic deformations.…”

“…On the other hand, we shall mean by h-∂ ∂-manifold for any h ∈ R \ {0}, a manifold that satisfies the h-∂ ∂-lemma (see definition 4.1) as defined in [2]. As mentioned above, it is already proved in [1] that the p-SKT property is open under holomorphic deformations on a ∂ ∂-manifold. The purpose of this section is the following Theorem 1.2.…”

“…DR (X, C) has been introduced, in [11], by requiring that the De Rham cohomology class {ω n−1 } ∈ H 2n−2 DR (X, C) be of type (n − 1, n − 1) for the complex structure J t of X t and are parametrised by (1). Following the pattern of the deformations of balanced Calabi-Yau ∂ ∂-manifold, we shall propose the following weakened notion:…”

“…Additionally, the strict weak positivity of Ω 0 implies the strict weak positivity of Ω t for t close to 0 (cf. [1], Lemma 4. Proof.…”