On astheno-Kähler metrics

Abstract

A Hermitian metric on a complex manifold of complex dimension n is called astheno-Kähler if its fundamental 2-form F satisfies the condition ∂∂F n−2 = 0. If n = 3, then the metric is strong KT, that is, F is ∂∂-closed. By using blow-ups and the twist construction, we construct simply connected astheno-Kähler manifolds of complex dimension n > 3. Moreover, we construct a family of astheno-Kähler (non-strong KT) 2-step nilmanifolds of complex dimension 4 and we study deformations of strong KT structures on nilmanifolds of complex dimension 3. Finally, we study the relation between the astheno-Kähler condition and the (locally) conformally balanced condition and we provide examples of locally conformally balanced astheno-Kähler metrics on T 2 -bundles over (non-Kähler) homogeneous complex surfaces.

A flow of conformally balanced metrics with Kähler fixed points

“…This statement can be viewed as a generalization to arbitrary dimensions of the arguments of [9]. It was proved in Matsuo-Takahashi [53], and extended in Fino-Tomassini [24] to the case of metrics which are both conformally balanced and astheno-Kähler. For easy reference, we state these results as a lemma, and include an alternative proof in this paper (see Corollary 1 in §3 for (ii) and (e) in §6 for (i)):…”

“…Another defining feature of the flow (1.1) is that it preserves the conformally balanced condition (1.2), and its stationary points are astheno-Kähler metrics (see §2.2 for definition). This implies that its stationary points are Kähler [24,53], so the convergence of the flow is closely related to a well-known question in non-Kähler geometry, namely when is a conformally balanced manifold actually Kähler. Also closely related is another fundamental question in non-Kähler geometry and algebraic-geometric stability conditions [16,47,77,78], namely when does a positive (p, p) cohomology class admit as representative the p-th power of a Kähler form.…”

Product of generalized p-Kähler manifolds

“…In the other cases this does not work: for instance, ∂∂ω 2 = 0 does not imply ∂∂ω = 0, as some examples in [12], (2.3) show.…”

On non-Kähler compact complex manifolds with balanced and astheno-Kähler metrics

“…Remark 2.5. The (real) nilmanifolds in (5) correspond to the Lie algebras g = h 2n+1 × R, where h 2n+1 is the (2n + 1)-dimensional Heisenberg algebra. Andrada, Barberis and Dotti proved in [2, Proposition 2.2] that every invariant complex structure J on these nilmanifolds is abelian, i.e.…”

“…If the Lee form is exact, then the Hermitian structure is conformally balanced. By [5,11] a conformally balanced SKT or astheno-Kähler metric whose Bismut connection has (restricted) holonomy contained in SU (n) is necessarily Kähler. Similar results for 1-st Gauduchon metrics are proved in [6].…”