Monte Carlo methods on compact complex manifolds using Bergman kernels

Numerical integration on manifolds is fundamental in many scientific fields, yet high-dimensional spaces pose severe challenges. Classical Monte Carlo methods are unbiased but slow, with convergence rates of N^{-1/2}. Quasi-Monte Carlo improves speed but sacrifices unbiasedness. Recent advances incorporate determinantal point processes (DPPs) for better point distributions, especially in Euclidean spaces. In complex geometry, Bergman kernels serve as projection operators with rich asymptotic properties, enabling structured sampling. Prior work on spheres and projective spaces demonstrated promising results, but a general framework on arbitrary compact complex manifolds was lacking. This paper aims to fill this gap by leveraging the geometric structure of Bergman kernels to design efficient, unbiased quadrature schemes with optimal convergence rates.

Achieving high-precision, unbiased numerical integration on complex manifolds remains difficult due to the geometric complexity and the need for well-distributed sampling points. Existing methods either lack theoretical guarantees of convergence speed or are computationally infeasible in high dimensions. The core challenge is to develop a sampling strategy that respects the manifold's complex structure, ensures low variance, and achieves optimal asymptotic error decay. Additionally, establishing a rigorous central limit theorem for such estimators in the geometric setting is non-trivial, especially when the kernel functions depend on the sample size and measure reweighting. Overcoming these issues is crucial for advancing computational methods in complex geometry and related fields.

The main innovation is integrating the Bergman kernel’s geometric properties with DPPs to produce a point process that yields optimal convergence rates for integral estimates. This involves: 1) constructing a DPP based on the Bergman kernel associated with a positive line bundle; 2) deriving a CLT for weighted linear statistics with varying test functions; 3) establishing an unbiased estimator through a reweighting scheme that maintains universality across measures. These steps collectively enable a high-dimensional, geometrically faithful, and computationally feasible quadrature method, surpassing previous Euclidean results and extending probabilistic numerical analysis into complex geometry.

• �� Select a compact complex manifold M with a positive holomorphic line bundle L, equipped with a Hermitian metric h and measure μ. • For large k, construct the space H0(M, L^k) and its orthonormal basis, leading to the Bergman kernel B(kϕ, μ). • Define a DPP with kernel B(kϕ, μ), generating points that respect the manifold’s complex structure. • Use Berman’s CLT to analyze the linear statistics of the point process, ensuring asymptotic normality. • Develop an unbiased estimator by normalizing the sum of function evaluations weighted by the inverse Bergman density. • Incorporate a reweighting function to adapt to different measures, maintaining universality. • Implement numerical algorithms for kernel evaluation and sampling, validating through simulations on the sphere and other manifolds.

Experiments focus on the Riemann sphere, comparing the proposed Bergman DPP-based estimator with classical Monte Carlo and QMC methods. The setup involves sampling points via Legendre DPP and randomized spiral algorithms, then estimating integrals of smooth functions. Metrics include mean squared error, convergence rate, and empirical distribution of fluctuations. Hyperparameters such as the kernel scaling and reweighting functions are tuned to observe their impact on accuracy. Results show that with N=1000 samples, errors drop below 10^{-4}, confirming the theoretical rate of N^{-1-2/d_{R}}. Additional tests verify the estimator’s robustness under measure reweighting and different kernel parameters, demonstrating broad applicability and stability.

The estimator achieves the optimal convergence rate of N^{-1-2/d_{R}}, with numerical errors in the sphere example below 10^{-4} at N=1000. The variance remains stable under measure reweighting, confirming universality. The CLT is validated by normality tests on the fluctuation distribution. The method outperforms traditional Monte Carlo by a factor of 2-3 in error decay speed, especially in high dimensions, and maintains unbiasedness across different measures. These results demonstrate the theoretical guarantees translate effectively into practical performance.

Applicable in quantum physics for path integrals, in complex geometry for curvature integrals, and in machine learning for high-dimensional data sampling. The method requires the manifold’s geometric data and the ability to compute Bergman kernels, making it suitable for problems where geometric fidelity and efficiency are critical. Its universality allows adaptation to various measures, broadening its utility in scientific computing and simulation tasks.

Dependence on the positivity of the line bundle and smoothness assumptions limits applicability to certain classes of manifolds. Computing Bergman kernels in high dimensions remains computationally intensive, posing scalability challenges. Extending the framework to non-compact or singular spaces requires further theoretical development. Practical implementation in complex geometries with intricate topology is still under exploration, necessitating more efficient algorithms.