In this paper, we propose a new randomized method for numerical integration on a compact complex manifold with respect to a continuous volume form. Taking for quadrature nodes a suitable determinantal point process, we build an unbiased Monte Carlo estimator of the integral of any Lipschitz function, and show that the estimator satisfies a central limit theorem, with a faster rate than under independent sampling. In particular, seeing a complex manifold of dimension $d$ as a real manifold of dimension $d_{\mathbb{R}}=2d$, the mean squared error for $N$ quadrature nodes decays as $N^{-1-2/d_{\mathbb{R}}}$; this is faster than previous DPP-based quadratures and reaches the optimal worst-case rate investigated by [Bakhvalov 1965] in Euclidean spaces. The determinantal point process we use is characterized by its kernel, which is the Bergman kernel of a holomorphic Hermitian line bundle, and we strongly build upon the work of Berman that led to the central limit theorem in [Berman, 2018].We provide numerical illustrations for the Riemann sphere.

翻译：本文提出了一种新的随机方法，用于在紧复流形上关于连续体积形式进行数值积分。通过选取合适的行列式点过程作为求积节点，我们构建了任意Lipschitz函数积分的无偏蒙特卡洛估计量，并证明该估计量满足中心极限定理，其收敛速度优于独立采样情形。特别地，将d维复流形视为d_R=2d维实流形时，N个求积节点的均方误差以N^{-1-2/d_R}速率衰减；这比以往基于DPP的求积方法更快，达到了[Bakhvalov 1965]在欧氏空间中研究的最优最坏情形收敛速率。所使用的行列式点过程由其核函数（即全纯Hermite线丛的Bergman核）刻画，我们主要基于Berman的工作——该工作导出了[Berman, 2018]中的中心极限定理。我们给出了Riemann球面上的数值示例。