On the Bergman kernel of polarized abelian varieties

TL;DR

The paper derives an explicit Bergman-kernel formula for positive line bundles on polarized abelian varieties, expressing $\rho_k(p)$ as a lattice sum with Gaussian weights and holonomy corrections: $\rho_k(p)=(\frac{k}{2\pi})^n\left(1+\sum_{v\in \Lambda\setminus\{0\}} e^{-\frac{k}{4}|v|_H^2}\cos(2\pi k\alpha_v(p))\right)$. The approach lifts the problem to $\mathbb{C}^n$, analyzes a flat cylinder and its higher-dimensional analogues, and then sums over the lattice to obtain a global formula reminiscent of Selberg-type trace sums. Key contributions include a rigidity result: if two polarizations share the same first Chern class and yield identical Bergman kernels for some $k$, then the corresponding $k$-th tensor powers are isomorphic; plus localization of the Bergman-density maxima/minima and an exponential off-diagonal decay bound $|K_k(x,y)|_h^k\le(\frac{k}{2\pi})^n\sum_{\gamma\in\mathfrak{G}_{x,y}} e^{-\frac{k}{4}\ell(\gamma)^2}$. These results provide a concrete, globally computable picture of Bergman kernels on abelian varieties and connect kernel data to holonomy and lattice geometry, with potential applications in geometric quantization and rigidity questions.

Abstract

We prove an explicit formula for the Bergman kernel of polarized abelian varieties. As applications, we show that if two positive line bundles represent the same first Chern class and have identical Bergman kernel functions for some tensor power, then the corresponding powers of the bundles are isomorphic. We also obtain localization results for the maxima and minima of the Bergman kernel function and establish exponential off-diagonal decay estimates.

Theorems & Definitions (30)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Conjecture 1.4
• Theorem 1.5
• Theorem 2.1: Poisson summation formula
• Theorem 3.1
• Lemma 3.2
• proof
• proof : Proof of Theorem \ref{['thm-cylinder-main']}