Abstract
We investigate the geometry of Hermitian manifolds endowed with a compact Lie group action by holomorphic isometries with principal orbits of codimension one. In particular, we focus on a special class of these manifolds constructed by following Bérard-Bergery which includes, among the others, the holomorphic line bundles on $\mathbb {C}\mathbb {P}^{m-1}$, the linear Hopf manifolds and the Hirzebruch surfaces. We characterize their invariant special Hermitian metrics, such as balanced, Kähler-like, pluriclosed, locally conformally Kähler, Vaisman and Gauduchon. Furthermore, we construct new examples of cohomogeneity one Hermitian metrics solving the second-Chern–Einstein equation and the constant Chern-scalar curvature equation.
Highlights
One of the most useful ways to construct concrete examples of Einstein metrics is by considering Riemannian manifolds with a large symmetry group, for example, homogeneous spaces and manifolds of cohomogeneity one, see e.g. [10, 56, 57] and references therein
The Calabi-Yau theorem assures the existence of Einstein metrics on compact complex Kahler manifolds with non-positive first Chern class [8, 59], and the existence of Kahler-Einstein metrics on Fano manifolds has been recently understood
The Einstein equation is reduced to a system of second-order ODEs, in both the spaces En and Mn, that can be integrated to get Einstein metrics which are Hermitian, see [9, Theoreme 1.10 and Theoreme 1.13]
Summary
One of the most useful ways to construct concrete examples of Einstein metrics is by considering Riemannian manifolds with a large symmetry group, for example, homogeneous spaces and manifolds of cohomogeneity one, see e.g. [10, 56, 57] and references therein. As another useful tool, the Calabi-Yau theorem assures the existence of Einstein metrics on compact complex Kahler manifolds with non-positive first Chern class [8, 59], and the existence of Kahler-Einstein metrics on Fano manifolds has been recently understood. The first non-homogeneous example of compact Riemannian Einstein manifold with positive scalar curvature has been provided by Page [45, 46] on CP2#CP2, and generalized by Berard-Bergery in [9] as follows. Let P be a compact Kahler-Einstein manifold with positive scalar curvature [1, 34, 35], for example, P = CP1 in case of the Page example. The only compact complex non-Kahler surface admitting second-Chern-Einstein metrics is the Hopf surface [24, Theorem 2], see [22]. Note that there still miss (if any) non-Kahler examples of second-Chern-Einstein metrics with negative Chern-scalar curvature on compact complex manifolds. 2-form associated to the metric, m denotes the complex dimension and θ is the Lee form of g [23], see Section
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