Convexity of the Bergman Kernels on Convex Domains

Abstract

Let $\Omega$ be a convex domain in ${\mathbb{C}}^n$ and $\phi$ a convex function on $\Omega$. We prove that $\log K_{\Omega ,\phi }(z)$ is a convex function (might be identically $-\infty$) on $\Omega$, where $K_{\Omega ,\phi }$ is the weighted Bergman kernel. When $\phi \equiv 0$, we prove a Brunn-Minkowski type inequality, which further implies that $K_{\Omega }(z{)}^{-\frac{1}{2n}}$ is a convex function if $\Omega$ is convex. Some necessary and sufficient conditions for strictly convexity are also obtained.

10 pages. A mistake in the proof of Corollary 1.3 in the previous version is corrected. Some discussions on the d-boundedness of the Bergman metric is added. Comments are welcome!