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  "body": "\n## Definition\n\nLet $\\Omega \\subseteq \\C^n$ be a bounded domain.\nLet $K(z, w)$ be its [[Bergman kernel|8RWDHW0CXC]], the reproducing kernel of the [[Bergman space|2D619WTPYC]] $A^2(\\Omega)$ of holomorphic functions in $L^2(\\Omega)$.\nThe *Bergman metric* of $\\Omega$ is the [[Hermitian metric|CK1DCPC3D3]]\n$$ds^2 = \\sum_{j,k=1}^n g_{j\\bar k}(z) \\, dz_j \\, d\\bar z_k, \\qquad g_{j\\bar k}(z) = \\frac{\\partial^2}{\\partial z_j \\partial \\bar z_k} \\log K(z, z) .$$\n\n## Remarks\n\nOn a bounded domain $K(z, z) \u003e 0$, so $\\log K(z, z)$ is defined and smooth.\nThe matrix $(g_{j\\bar k})$ is positive definite at every point; the bounded functions $1$ and $z_j$ in $A^2(\\Omega)$ give this positivity.\nIf $F \\colon \\Omega_1 \\to \\Omega_2$ is a biholomorphism, then $K_1(z, w) = \\det F'(z) \\, K_2(F(z), F(w)) \\, \\overline{\\det F'(w)}$, and the factors $\\det F'$ disappear under $\\partial\\bar\\partial \\log$, so $F$ is an isometry.\nExample: for the unit disc, $K(z, w) = \\frac{1}{\\pi (1 - z \\bar w)^2}$, so $\\log K(z, z) = -\\log \\pi - 2 \\log(1 - \\abs{z}^2)$ and $g_{1\\bar 1} = \\frac{2}{(1 - \\abs{z}^2)^2}$.\nExample: for the unit ball of $\\C^n$, the Bergman metric is $n + 1$ times the metric with potential $-\\log(1 - \\abs{z}^2)$.\nNon-example: for $\\Omega = \\C^n$ the Bergman space is $\\set{0}$, so $K \\equiv 0$ and the definition does not apply.\nThe Bergman metric is a Kähler metric, since it has a global potential.\n",
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