Bergman Metrics Overview

• Bergman metrics are canonical Kähler metrics defined via Bergman kernels in bounded domains, serving as biholomorphically invariant measures of complex structure.
• They exhibit rigidity properties, exemplified by the unit ball case with constant negative curvature, confirming key conjectures in complex geometry.
• Extensions include weighted and random variants that provide statistical interpretations and finite-dimensional approximations for quantum limit structures.

A Bergman metric is a canonical Kähler metric intrinsically associated with a domain or complex manifold via the Bergman kernel. The rich structure of Bergman metrics, their analytic and geometric properties, and their connections with information geometry and metric equivalence underpin substantial developments in complex analysis, several complex variables, complex differential geometry, and mathematical physics. This article provides a comprehensive view of Bergman metrics, their explicit realizations, rigidity phenomena, weighted and random variants, statistical interpretations, and recent advances.

1. Definition and Construction of the Bergman Metric

Let $D \subset \mathbb{C}^n$ be a bounded domain. The Hilbert space $A^2(D)$ consists of square-integrable holomorphic functions with respect to Lebesgue measure. Any orthonormal basis %%%%2%%%% defines the Bergman kernel: $$K_D(z, w) = \sum_{j=0}^\infty s_j(z)\overline{s_j(w)}.$$ The diagonal $K_D(z,z) > 0$ is strictly positive. The Bergman metric is the Kähler metric whose potential is $\log K_D(z, z)$: $$g_{i\bar j}(z) = \frac{\partial^2}{\partial z_i \partial \bar z_j} \log K_D(z,z),$$ with associated (1,1)-form

$$\omega_B = i\sum_{i,j=1}^n g_{i\bar j}(z)\,dz_i\wedge d\bar z_j.$$

This metric is biholomorphically invariant and is uniquely determined by the complex structure and the underlying measure on the domain (Yum, 2024).

Model Example: Unit Ball

For $\mathbb{B}^n = \{|z|<1\}$,

$$K_{\mathbb{B}^n}(z, w) = \frac{n!}{\pi^n} (1 - \langle z, w\rangle)^{-n-1}$$

and the Bergman metric has constant negative holomorphic sectional curvature $-4$ (Ganguly et al., 2022).

2. Canonical and Weighted Bergman Metrics

Weighted Bergman spaces generalize classical constructions by introducing a measurable weight $w(z)>0$, yielding a weighted kernel $K_w(z, \bar w)$ and associated metric. Crucially, under an invariant weight assignment $\mathcal{M}$, weighted Bergman metrics remain biholomorphically invariant: $$\omega_{\mu_D} = F^*\omega_{\mu_{D'}} \quad \text{ for } F: D \to D'$$ when the weights transform as $\mu_{D'}\circ F = |h_F|^2 \mu_D$ (Yoo, 2024).

Important examples include:

• Tian's sequence: Assigning powers of the determinant of the Kähler-Einstein metric.
• Tsuji's dynamical kernel: Inductive constructions where each weight is reciprocal to the preceding Bergman kernel.

On uniform squeezing domains, the Tian-Yau-Zelditch expansion ensures uniform convergence of weighted Bergman metrics to the Kähler-Einstein metric (Yoo, 2024).

3. Rigidity and Curvature Phenomena

Bergman metrics exhibit rigidity: if on a bounded domain the Bergman metric is Kähler-Einstein, then the domain is biholomorphic to the ball. This is a central result answering Yau's question and confirming Cheng's rigidity conjecture in all dimensions (Huang et al., 2016, Savale et al., 2023, Ganguly et al., 2022).

• Cheng's Conjecture: The Bergman metric is Kähler-Einstein if and only if the domain is biholomorphic to the ball.
• Ball Quotients: For finite $\Gamma$ and two-dimensional Stein spaces $B^2/\Gamma$, the Bergman metric is Kähler-Einstein if and only if $\Gamma$ is trivial (Ganguly et al., 2022).
• Cartan-Hartogs Domains: For $M_{\Omega,\mu}$, the following are equivalent (Loi et al., 7 Oct 2025):
  • $M_{\Omega,\mu} \simeq \mathbb{B}^{n+1}$,
  • The Bergman metric is a Kähler-Ricci soliton,
  • After rescaling, the Bergman dual is finitely projectively induced.

The exceptional status of the unit ball is reinforced in settings such as Cartan-Hartogs, Hartogs domains over homogeneous bases, and egg domains over irreducible symmetric bases (Palmieri, 20 Oct 2025).

4. Statistical and Information-Geometric Aspects

An embedding $\Phi: D \to \mathcal{P}(D)$, where $\mathcal{P}(D)$ is the space of probability measures, allows interpreting the Bergman metric as the pullback of the Fisher information metric: $\Phi^* g_F = g_B,$ with

$P(z,\xi) = \frac{|K(z, \xi)|^2}{K(z,z)}$

the Poisson–Bergman kernel, yielding a statistical model structure on $D$ (Cho et al., 2023).

A statistical curvature formula relates holomorphic sectional curvature to the covariance of $\partial^2 P / P$ (Cho et al., 2023): $$R_{\alpha\bar\alpha\alpha\bar\alpha}(z_0) = 2 - \mathbb{E}_{z_0}\left[ \frac{|\partial_{z_\alpha}\partial_{z_\alpha}P(z_0,\xi)|^2}{P(z_0,\xi)^2} \right].$$ Calabi's diastasis becomes a Bregman-type divergence, enabling analysis of Fréchet means and associated central limit behavior (Cho et al., 2023).

Rigidities utilizing sufficient statistics imply that, under mild regularity, Bergman local isometries must be biholomorphisms (Yum, 2024), and the Schwarz lemma for Bergman metrics can be deduced from Cauchy-Schwarz inequalities in this probabilistic context (Seo et al., 30 Dec 2025).

5. Bergman Metrics as Approximations and Quantum Limit Structures

Bergman metrics also serve as finite-dimensional approximations to infinite-dimensional spaces of Kähler metrics. On polarized Kähler manifolds $(M, L, g, x)$, the Kodaira embedding yields: $$\omega_k = \frac{i}{k} \partial\bar\partial \log K_k(z, z)$$ where $K_k$ is the Bergman kernel for $L^k$. Uniform $C^{1,\alpha}$ convergence holds as $k\to\infty$ at optimal rates up to $\mathcal{O}(k^{-2}|\log k|^\alpha)$ (Zhou, 2021). For symplectic and almost-Kähler manifolds, analogous convergence results hold for Bergman metrics constructed from low-energy eigensections of the Bochner Laplacian, with speed $1/p^2$ (Lu et al., 2017).

6. Random Bergman Metrics and Stability

Randomization of Bergman metrics by integrating over finite-dimensional symmetric spaces ($SL(N_k,\mathbb{C})/SU(N_k)$) with Haar measure leads to a family of random metrics, with important physical and geometric interpretations (Ferrari et al., 2011, Klevtsov et al., 2014). Partition functions for random Bergman metrics reveal stability invariants: the critical coupling constant $\gamma_k^{\rm crit}$ governs integrability and connects to geodesic stability and large deviations, with explicit expressions in genus-one cases $\gamma_k^{\rm crit} = k-h$ (Klevtsov et al., 2014).

Expectation values of random Bergman metrics converge to the background Kähler metric, and higher-point functions display universality and contact phenomena related to the structure of the underlying space of sections.

7. Relationships to Other Invariant Metrics and Open Problems

On bounded convex domains, the Bergman metric, Carathéodory-Reiffen metric, and complete Kähler-Einstein metric of negative scalar curvature are uniformly equivalent but not proportional except on the unit ball $\mathbb{B}^n$ (Cho, 2018). For irreducible bounded symmetric domains in their Harish-Chandra realization, explicit comparison with the generalized Hilbert metric and Finsler norm quantifies distinctions (Falbel et al., 2024):

Metric Type	Infinitesimal norm at 0
Carathéodory–Kobayashi	$\max_i \sigma_i$
Hilbert (FGW)	$2\sum_i \sigma_i$
Bergman	$\sqrt{\gamma \sum_i \sigma_i^2}$

Open questions concern the maximal domains where the Bergman dual metric is defined, conjectures on positivity of holomorphic sectional curvature for Stein manifolds, and the explicit metric classification in the presence of singularities (Loi et al., 7 Oct 2025, Huang et al., 2023).

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References:

• (Loi et al., 7 Oct 2025) On the Bergman metric of Cartan-Hartogs domains
• (Palmieri, 20 Oct 2025) Bergman metrics induced by the ball
• (Yum, 2024) Bergman local isometries are biholomorphisms
• (Savale et al., 2023) Kähler-Einstein Bergman metrics on pseudoconvex domains of dimension two
• (Cho et al., 2023) Statistical Bergman geometry
• (Klevtsov et al., 2014) Stability and integration over Bergman metrics
• (Huang et al., 2016) Bergman-Einstein metrics, hyperbolic metrics and Stein spaces with spherical boundaries
• (Cho, 2018) Invariant metrics on the Complex ellipsoid
• (Yoo, 2024) Invariant weighted Bergman metrics on domains
• (Huang et al., 2023) Bergman metrics as pull-backs of the Fubini-Study metric
• (Seo et al., 30 Dec 2025) On the Schwarz Lemma for Bergman metrics of bounded domains
• (Zhou, 2021) On the convergence rate of Bergman metrics
• (Lu et al., 2017) Optimal convergence speed of Bergman metrics on symplectic manifolds
• (Ferrari et al., 2011) Simple matrix models for random Bergman metrics
• (Ganguly et al., 2022) Bergman-Einstein metrics on two-dimensional Stein spaces
• (Falbel et al., 2024) A Hilbert metric for bounded symmetric domains
• (Cho et al., 2021) Equivalence of Invariant metrics via Bergman kernel on complete noncompact Kähler manifolds