  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.
