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- W3045804938 abstract "Let $(M,h)$ be a Hermitian manifold and $psi$ a smooth weight function on $M$. The $partial$-complex on weighted Bergman spaces $A^2_{(p,0)}(M,h, e^{-psi})$ of holomorphic $(p,0)$-forms was recently studied in [[10] and [9]. It was shown that if $h$ is Kahler and a suitable density condition holds, the $partial$-complex exhibits an interesting holomorphicity/duality property when $(barpartialpsi)^{sharp}$ is holomorphic (i.e., when the real gradient field $mathrm{grad}_hpsi$ is a real holomorphic vector field). For general Hermitian metrics this property does not hold without the holomorphicity of the torsion tensor $T_p{}^{rs}$. In this paper, we investigate the existence of real-valued weight functions with real holomorphic gradient fields on Kahler and conformally Kahler manifolds and their relationship to the $partial$-complex on weighted Bergman spaces. For Kahler metrics with multi-radial potential functions on $mathbb C^n$ we determine all multi-radial weight functions with real holomorphic gradient fields. For conformally Kahler metrics on complex space forms we first identify the metrics having holomorphic torsion leading to several interesting examples such as the Hopf manifold $mathbb{S}^{2n-1} times mathbb{S}^1$, and the hyperbolic metric on the unit ball. For some of these metrics, we further determine weight functions $psi$ with real holomorphic gradient fields. They provide a wealth of triples $(M,h,e^{-psi})$ of Hermitian non-Kahler manifolds with weights for which the $partial$-complex exhibits the aforementioned holomorphicity/duality property. Among these examples, we study in detail the $partial$-complex on the unit ball with the half hyperbolic metric and derive a new estimate for the $partial$-equation." @default.
- W3045804938 created "2020-08-03" @default.
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- W3045804938 date "2020-07-29" @default.
- W3045804938 modified "2023-09-27" @default.
- W3045804938 title "The $partial$-Operator and Real Holomorphic Vector Fields" @default.
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