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Register a new userRelations between the Kahler cone and the balanced cone of a Kahler manifold
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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.
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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.
For a compact Lie group G we define a regularized version of the Dolbeault cohomology of a G-equivariant holomorphic vector bundles over non-compact Kahler manifolds. The new cohomology is infinite-dimensional, but as a representation of G it decomposes into a sum of irreducible components, each of which appears in it with finite multiplicity. Thus equivariant Betti numbers are well defined. We study various properties of the new cohomology and prove that it satisfies a Kodaira-type vanishing theorem.
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 $S^3times S^3$.
We review the information geometry of linear systems and its application to Bayesian inference, and the simplification available in the Kahler manifold case. We find conditions for the information geometry of linear systems to be Kahler, and the relation of the Kahler potential to information geometric quantities such as $alpha $-divergence, information distance and the dual $alpha $-connection structure. The Kahler structure simplifies the calculation of the metric tensor, connection, Ricci tensor and scalar curvature, and the $alpha $-generalization of the geometric objects. The Laplace--Beltrami operator is also simplified in the Kahler geometry. One of the goals in information geometry is the construction of Bayesian priors outperforming the Jeffreys prior, which we use to demonstrate the utility of the Kahler structure.
We give an explicit formula for the $L^2$ analytic torsion of the finite metric cone over an oriented compact connected Riemannian manifold. We provide an interpretation of the different factors appearing in this formula. We prove that the analytic torsion of the cone is the finite part of the limit obtained collapsing one of the boundaries, of the ratio of the analytic torsion of the frustum to a regularising factor. We show that the regularising factor comes from the set of the non square integrable eigenfunctions of the Laplace Beltrami operator on the cone.
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