Deformations of Strong Kähler with torsion metrics

Abstract

Existence of strong Kähler with torsion metrics, shortly SKT metrics, on complex manifolds has been shown to be unstable under small deformations. We find necessary conditions under which the property of being SKT is stable for a smooth curve of Hermitian metrics {ω t } t which equals a fixed SKT metric ω for t = 0, along a differentiable family of complex manifolds {M t } t .

Deformations of balanced metrics

“…In this paper, following a similar approach to [23] and [27], we prove necessary conditions of cohomological type for the existence of a curve of astheno-Kähler metrics starting from a fixed astheno Kähler metric on a compact complex manifold. More specifically, if e ιϕ ι ϕ denotes the extension map introduced in [25] and recalled in section 2, and ι ψ is the contraction by a (0, 1)vector form with values in a holomorphic vector bundle, as we recall in section 3, we have obtained the following results.…”

Deformations of balanced metrics

“…Note that this method of computing ∂ t and ∂ t does not require any information on the complex coordinates of the deformed complex manifold. Furthermore, we point out that this method could be applied to find necessary conditions to the existence smooth families of special Hermitian metrics other than balanced ones, for example SKT metrics, as done in [15]. This note is organized as follows.…”