The Bergman metric form on a bounded domain

Assume the Axiom of Countable Choice ${\mathrm{A}\mathrm{C}}_{\omega }$ (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)), let $m\ge 1$, and let $\mathrm{\Omega }\subseteq {\mathbb{C}}^m$ be a bounded domain, meaning a nonempty connected open set. Use the first-variable-linear Bergman kernel $K_{\mathrm{\Omega }}$ of The Bergman space $A^2(\mathrm{\Omega })$ and the Bergman kernel. By Smoothness of the Bergman kernel and positivity of its diagonal on bounded domains, $u_{\mathrm{\Omega }}(z):=\log K_{\mathrm{\Omega }}(z,z)$ is a real-valued $C^{\mathrm{\infty }}$ function on $\mathrm{\Omega }$.

Using the one-based coordinate aliases for the library's zero-based coordinates from The Levi form and strict plurisubharmonicity, define the Bergman metric form at $z\in \mathrm{\Omega }$ by the Hermitian form $g_{\mathrm{\Omega }}(z)(X,Y):={\sum }_{j,k=1}^m\frac{{\mathrm{\partial }}^2{u}_{\mathrm{\Omega }}}{\mathrm{\partial }{z}_j\mathrm{\partial }{\bar{z}}_k}(z)X_j\overline{Y_k},X,Y\in {\mathbb{C}}^m\mathrm{.}$ Because $u_{\mathrm{\Omega }}$ is real-valued and $C^2$, expanding the Wirtinger operators and commuting real mixed partials gives $\overline{g_{j\stackrel{ˉ}{k}}}=g_{k\bar{j}}$ (Wirtinger operators in ${\mathbb{C}}^m$, Clairaut--Schwarz theorem for continuous second partial derivatives), so this form is Hermitian. Its associated quadratic form is $B_{\mathrm{\Omega }}^2(z;X):=g_{\mathrm{\Omega }}(z)(X,X),$ the Levi form of $u_{\mathrm{\Omega }}$ at $z$; write its matrix as $g_{\mathrm{\Omega }}(z)=(g_{j\bar{k}}(z))$. The pair $(\mathrm{\Omega },g_{\mathrm{\Omega }})$ is the Bergman (pseudo-)metric. This definition does not assert positive definiteness; the separate positivity theorem on this page proves it for bounded domains.

The ${\mathrm{A}\mathrm{C}}_{\omega }$ assumption is inherited through the Bergman Hilbert-space/Riesz construction and its smoothness/positive-diagonal supplier. The definition itself makes no additional selections and uses no full Axiom of Choice.