  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 K\"ahler and a suitable density condition holds,
the $\partial$-complex exhibits an interesting holomorphicity/duality property
when $(\bar\partial\psi)^{\sharp}$ is holomorphic (i.e., when the real gradient
field $\mathrm{grad}_h\psi$ 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 K\"ahler and conformally K\"ahler
manifolds and their relationship to the $\partial$-complex on weighted Bergman
spaces. For K\"ahler metrics with multi-radial potential functions on $\mathbb
C^n$ we determine all multi-radial weight functions with real holomorphic
gradient fields. For conformally K\"ahler 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 "half" 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-K\"ahler 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.
