On cohomogeneity one Hermitian non-K\"ahler metrics

TL;DR

This paper studies special Hermitian metrics on cohomogeneity one manifolds with Lie group symmetries, characterizing known types and constructing new examples satisfying specific curvature equations.

Contribution

It characterizes invariant special Hermitian metrics on a class of cohomogeneity one manifolds and constructs new solutions to curvature equations.

Findings

Characterization of balanced, K"ahler-like, pluriclosed, locally conformally K"ahler, Vaisman, Gauduchon metrics.

Construction of new cohomogeneity one Hermitian metrics solving second-Chern-Einstein and constant Chern-scalar curvature equations.

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 B\'erard-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, K\"ahler-like, pluriclosed, locally conformally K\"ahler, Vaisman, Gauduchon. Furthermore, we construct new examples of cohomogeneity one Hermitian metrics solving the second-Chern-Einstein equation and the constant Chern-scalar curvature equation.