Central Limit theorem for toric \kahler manifolds

Abstract

Associated to the Bergman kernels of a polarized toric \kahler manifold $(M,\omega ,L,h)$ are sequences of measures $\{{\mu }_k^z{\}}_{k=1}^{\infty }$ parametrized by the points $z\in M$. For each $z$ in the open orbit, we prove a central limit theorem for ${\mu }_k^z$. The center of mass of ${\mu }_k^z$ is the image of $z$ under the moment map; after re-centering at $0$ and dilating by $\sqrt{k}$, the re-normalized measure tends to a centered Gaussian whose variance is the Hessian of the \kahler potential at $z$. We further give a remainder estimate of Berry-Esseen type. The sequence $\{{\mu }_k^z\}$ is generally not a sequence of convolution powers and the proofs only involve \kahler analysis.

Cited by in corpus (1)

• Entropy of Bergman measures of a toric Kaehler manifold