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We study the existence of three classes of Hermitian metrics on certain types of compact complex manifolds. More precisely, we consider balanced, SKT and astheno-Kahler metrics. We prove that the twistor spaces of compact hyperkahler and negative quaternionic-Kahler manifolds do not admit astheno-Kahler metrics. Then we provide examples of astheno-Kahler structures on toric bundles over Kahler manifolds. In particular, we find examples of compact complex non-Kahler manifolds which admit a balanced and an astheno-Kahler metrics, thus answering to a question in [52] (see also [24]). One of these examples is simply connected. We also show that the Lie groups $SU(3)$ and $G_2$ admit SKT and astheno-Kahler metrics, which are different. Furthermore, we investigate the existence of balanced metrics on compact complex homogeneous spaces with an invariant volume form, showing in particular that if a compact complex homogeneous space $M$ with invariant volume admits a balanced metric, then its first Chern class $c_1(M)$ does not vanish. Finally we characterize Wang C-spaces admitting SKT metrics.
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We construct balanced metrics on the family of non-Kahler Calabi-Yau threefolds that are obtained by smoothing after contracting $(-1,-1)$-rational curves on Kahler Calabi-Yau threefold. As an application, we construct balanced metrics on complex manifolds diffeomorphic to connected sum of $kgeq 2$ copies of $S^3times S^3$.
In this paper, we consider a natural map from the Kahler cone to the balanced cone of a Kahler manifold. We study its injectivity and surjecticity. We also give an analytic characterization theorem on a nef class being Kahler.
We work in both the complex and in the para-complex categories and examine (para)-Kahler Weyl structures in both the geometric and in the algebraic settings. The higher dimensional setting is quite restrictive. We show that any (para)-Kaehler Weyl algebraic curvature tensor is in fact Riemannian in dimension at least 6; this yields as a geometric consequence that any (para)-Kaehler Weyl geometric structure is trivial if the dimension is at least 6. By contrast, the 4 dimensional setting is, as always, rather special as it turns out that there are (para)-Kaehler Weyl algebraic curvature tensors which are not Riemannian in dimension 4. Since every (para)-Kaehler Weyl algebraic curvature tensor is geometrically realizable and since every 4 dimensional Hermitian manifold admits a unique (para)-Kaehler Weyl structure, there are also non-trivial 4 dimensional Hermitian (para)-Kaehler Weyl manifolds.
We give a moment map interpretation of some relatively balanced metrics. As an application, we extend a result of S. K. Donaldson on constant scalar curvature Kahler metrics to the case of extremal metrics. Namely, we show that a given extremal metric is the limit of some specific relatively balanced metrics. As a corollary, we recover uniqueness and splitting results for extremal metrics in the polarized case.
We study balanced Hermitian structures on almost abelian Lie algebras, i.e. on Lie algebras with a codimension-one abelian ideal. In particular, we classify 6-dimensional almost abelian Lie algebras which carry a balanced structure. It has been conjectured by A. Fino and L. Vezzoni that a compact complex manifold admitting both a balanced metric and a SKT metric necessarily has a Kahler metric: we prove this conjecture for compact almost abelian solvmanifolds with left-invariant complex structures. Moreover, we investigate the behaviour of the flow of balanced metrics introduced by L. Bedulli and L. Vezzoni and of the anomaly flow of D. H. Phong, S. Picard and X. Zhang on almost abelian Lie groups. In particular, we show that the anomaly flow preserves the balanced condition and that locally conformally Kahler metrics are fixed points.
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