We can define a harmonic function on an open subset $ W $ of $ \mathbb{C}^{n} $. Let $ z_1, \cdots, z_n $ be the coordinates for $ \mathbb{R}^{n} $, where $ z_i=x_i+iy_i $. We have a multi-complex-variable real-valued function $ u(z_1,\cdots, z_n) $, if $$ \sum\limits_{i=1}^{n}\frac{\partial^{2}u}{\partial x_i^{2}}+\frac{\partial^{2}u}{\partial y_i^{2}}=0 $$ then we say $ u $ is a harmonic function on $ W $.
My question is: is it possible to define harmonic functions on complex manifolds, not just in $ \mathbb{C}^{n} $.
By the definition of complex manifolds, it only guarantees the maps between $ \mathbb{C}^{n} $ and $ U_i $ are holomorphic, where $ \{U_i\} $ is an open covering of the complex manifold. So what we have to prove in order to show that the definition of harmonic functions on complex manifolds(if possible) is that it is irrelevant to the change of coordinates, i.e.:
Given a harmonic function $ u(z_1,\cdots,z_n): \mathbb{C}^{n}\to\mathbb{R} $ on a complex manifold(Suppose we have successfully defined one.). For every biholomorphic function $ h(z_1,\cdots, z_n): \mathbb{C}^{n}\to\mathbb{C}^{n} $, then $ u(h(z_1,\cdots, z_n)): \mathbb{C}^{n}\to\mathbb{R} $ is also harmonic function on the same complex manifold.
Every complex manifold $M$ admits not only a Riemannian metric but a Hermitian metric, which is a Riemannian metric that preserves the complex structure; a complex manifold endowed with the Hermitian metric is called a Hermitian manifold. In general, though, a Hermitian manifold $M$ comes equipped with not one but three natural analogues of the Laplacian, namely, $$ \Delta_d := dd^\ast + d^\ast d, \quad \Delta_{\partial} := \partial\partial^\ast + \partial^\ast \partial, \quad \Delta_{\bar{\partial}} := \bar{\partial}\bar{\partial}^\ast + \bar{\partial}^\ast\bar{\partial}, $$ where for any differentiable function $f$, $$ df = \partial f + \bar{\partial} f, \quad \partial f := \sum_{k=1}^{\dim M} \frac{\partial f}{\partial z^k}dz^k, \quad \bar{\partial} f := \sum_{k=1}^{\dim M} \frac{\partial f}{\partial \bar{z}^k}d\bar{z}^k, $$ but their relationship can be a bit subtle. However, if $M$ is a Kähler manifold, then $$ \Delta_d = 2\Delta_\partial = 2\Delta_{\bar{\partial}}, $$ in which case, all three Laplacians give rise to the same notion of harmonic functions, and, more generally, harmonic differential forms of various (bi)degrees, and hence to the powerful mathematical framework of Hodge theory.