Statistical Bergman Geometry Insights

• Statistical Bergman Geometry is the study of recasting classical Bergman metrics into a statistical framework by embedding complex domains into spaces of smooth probability densities.
• It links the Fisher information metric to the Bergman metric, yielding curvature formulas, rigidity results for holomorphic maps, and asymptotic properties of Fréchet means.
• In the semiclassical Kähler setting, partial Bergman kernels generate determinantal point processes whose limit laws connect spectral fluctuations to intrinsic Sobolev norms.

Statistical Bergman Geometry studies Bergman-geometric structures by recasting them in statistical and probabilistic terms. For bounded domains $\Omega\subset\mathbb{C}^n$, the central construction is an embedding of $\Omega$ into a space of probability distributions via the Poisson–Bergman kernel, with the pullback of the Fisher information metric coinciding with the Bergman metric; within this framework one obtains curvature formulas, rigidity results for proper holomorphic maps, and asymptotic theory for the Fréchet sample mean associated with Calabi’s diastasis (Cho et al., 2023). In a semiclassical Kähler setting, related statistical phenomena arise from equivariant and partial Bergman kernels, which define determinantal point processes whose linear statistics satisfy a law of large numbers and a central limit theorem, with variance determined by a bulk $H^1$ term and a boundary $H^{1/2}$ term (Ioos, 25 Nov 2025).

1. Statistical model induced by the Bergman kernel

For a bounded domain $\Omega\subset\mathbb{C}^n$, let $A^2(\Omega)$ denote the Hilbert space of square-integrable holomorphic functions with inner product

$$\langle f,g\rangle=\int_\Omega f(z)\overline{g(z)}\,dV(z).$$

Its reproducing kernel is the Bergman kernel

$B(z,w)=\sum_{j=0}^\infty s_j(z)\overline{s_j(w)},$

where $\{s_j\}$ is any orthonormal basis of $A^2(\Omega)$, and $\Omega$0 for all $\Omega$1 (Cho et al., 2023).

The associated Poisson–Bergman, or Berezin, kernel is

$\Omega$2

For each fixed $\Omega$3, one has $\Omega$4 and

$\Omega$5

by the reproducing property. This yields the statistical embedding

$\Omega$6

where $\Omega$7 and $\Omega$8 denotes the manifold of all smooth probability densities on $\Omega$9.

The basic regularity properties are part of the construction. Because $H^1$0 is bounded and $H^1$1, for each $H^1$2 the density $H^1$3 is positive and $H^1$4 in $H^1$5; its support is all of $H^1$6 because zeros of $H^1$7 occur only on a complex subvariety of measure zero; and the map $H^1$8 is $H^1$9, with normalization differentiable under the integral sign. In this way, a complex domain is treated as a finite-dimensional statistical model sitting inside an infinite-dimensional manifold of probability densities.

2. Fisher information and the Bergman metric

On $H^{1/2}$0, the Fisher information metric is defined as follows. At $H^{1/2}$1, for tangent vectors $H^{1/2}$2 with $H^{1/2}$3,

$H^{1/2}$4

Restricting to the finite-dimensional submanifold $H^{1/2}$5, one identifies

$H^{1/2}$6

Since $H^{1/2}$7 and similarly for $H^{1/2}$8,

$H^{1/2}$9

Using the identity

$\Omega\subset\mathbb{C}^n$0

and the fact that $\Omega\subset\mathbb{C}^n$1 is independent of $\Omega\subset\mathbb{C}^n$2, integration gives

$\Omega\subset\mathbb{C}^n$3

Hence

$\Omega\subset\mathbb{C}^n$4

where

$\Omega\subset\mathbb{C}^n$5

This is the Burbea–Rao theorem in the form used here (Cho et al., 2023).

The significance of this identity is structural rather than merely formal. It identifies the Bergman metric as a Fisher metric pulled back from a statistical ambient space, so the domain $\Omega\subset\mathbb{C}^n$6 may be regarded as a statistical model whose intrinsic Riemannian geometry is exactly the classical Bergman geometry.

3. Curvature formulas in statistical form

Once the Bergman metric is realized as a pullback of the Fisher information metric, its curvature can be expressed through expectations of score-type quantities. In holomorphic normal coordinates at $\Omega\subset\mathbb{C}^n$7, where $\Omega\subset\mathbb{C}^n$8 and $\Omega\subset\mathbb{C}^n$9, the relevant holomorphic sectional curvature satisfies

$A^2(\Omega)$0

Because $A^2(\Omega)$1 in these coordinates,

$A^2(\Omega)$2

(Cho et al., 2023).

Introducing the random variable

$A^2(\Omega)$3

the curvature can be rewritten as

$A^2(\Omega)$4

In this formulation, the deviation of the Bergman holomorphic sectional curvature from the upper bound $A^2(\Omega)$5 is exactly the variance of the second-order score function. More generally, for two real tangent vectors $A^2(\Omega)$6, one obtains a similar covariance expression involving appropriate score functions along $A^2(\Omega)$7 and $A^2(\Omega)$8.

This rephrasing is one of the defining features of the subject. Curvature is no longer expressed only through differential-geometric tensors built from the Bergman kernel; it is also encoded by expectations and covariances in the associated statistical model.

4. Rigidity of proper holomorphic maps

Let $A^2(\Omega)$9 be bounded domains in $$\langle f,g\rangle=\int_\Omega f(z)\overline{g(z)}\,dV(z).$$0 with Bergman metrics $$\langle f,g\rangle=\int_\Omega f(z)\overline{g(z)}\,dV(z).$$1. Any measurable map $$\langle f,g\rangle=\int_\Omega f(z)\overline{g(z)}\,dV(z).$$2 induces a push-forward

$$\langle f,g\rangle=\int_\Omega f(z)\overline{g(z)}\,dV(z).$$3

By the general monotonicity of the Fisher metric under push-forward,

$$\langle f,g\rangle=\int_\Omega f(z)\overline{g(z)}\,dV(z).$$4

Equality is the information-geometric condition of sufficiency, or local information isometry:

$$\langle f,g\rangle=\int_\Omega f(z)\overline{g(z)}\,dV(z).$$5

The general theorem quoted in this framework states that

$$\langle f,g\rangle=\int_\Omega f(z)\overline{g(z)}\,dV(z).$$6

If, in addition, $$\langle f,g\rangle=\int_\Omega f(z)\overline{g(z)}\,dV(z).$$7 is proper holomorphic, then Remmert–Bell theory gives that $$\langle f,g\rangle=\int_\Omega f(z)\overline{g(z)}\,dV(z).$$8 is an $$\langle f,g\rangle=\int_\Omega f(z)\overline{g(z)}\,dV(z).$$9-sheeted covering outside a measure-zero branch locus, and a Cauchy–Schwarz / Bell–transformation rule argument shows that equality in Fisher monotonicity forces $B(z,w)=\sum_{j=0}^\infty s_j(z)\overline{s_j(w)},$0; therefore $B(z,w)=\sum_{j=0}^\infty s_j(z)\overline{s_j(w)},$1 is biholomorphic (Cho et al., 2023).

The resulting rigidity theorem is precise: if $B(z,w)=\sum_{j=0}^\infty s_j(z)\overline{s_j(w)},$2 is a proper holomorphic map and the induced push-forward preserves the Fisher information metrics on the embedded statistical models, then $B(z,w)=\sum_{j=0}^\infty s_j(z)\overline{s_j(w)},$3 must be a biholomorphism. The conclusion does not apply to arbitrary holomorphic maps without the properness and metric-preservation hypotheses; the force of the result lies in this exact conjunction of analytic and information-geometric assumptions.

5. Calabi’s diastasis and Fréchet mean asymptotics

For the Bergman metric, Calabi’s diastasis is

$B(z,w)=\sum_{j=0}^\infty s_j(z)\overline{s_j(w)},$4

Fix $B(z,w)=\sum_{j=0}^\infty s_j(z)\overline{s_j(w)},$5 and draw i.i.d. samples $B(z,w)=\sum_{j=0}^\infty s_j(z)\overline{s_j(w)},$6 from the probability distribution

$B(z,w)=\sum_{j=0}^\infty s_j(z)\overline{s_j(w)},$7

The Fréchet sample mean is defined by

$B(z,w)=\sum_{j=0}^\infty s_j(z)\overline{s_j(w)},$8

Under the regularity assumptions used in the construction of the model, one has existence and consistency,

$B(z,w)=\sum_{j=0}^\infty s_j(z)\overline{s_j(w)},$9

and a central limit theorem,

$\{s_j\}$0

Thus the asymptotic covariance of the estimator is exactly the inverse Bergman metric at the true parameter (Cho et al., 2023).

The asymptotic statement mirrors the Fisher-information/Cramér–Rao paradigm, but it is realized here through Calabi’s diastasis on a complex domain endowed with its Bergman geometry. This gives a statistical estimation-theoretic interpretation to the inverse Bergman metric and places Fréchet means within the same geometric framework as the metric and curvature formulas.

6. Semiclassical determinantal processes and statistical Bergman geometry

A second, semiclassical formulation appears for a prequantized Kähler manifold $\{s_j\}$1 of complex dimension $\{s_j\}$2 endowed with a Hermitian line bundle $\{s_j\}$3 satisfying

$\{s_j\}$4

assuming bounded geometry at infinity and a compatible Hamiltonian $\{s_j\}$5-action with Kostant moment map $\{s_j\}$6, proper and bounded below. The induced $\{s_j\}$7-action on $\{s_j\}$8 splits it as

$\{s_j\}$9

If $A^2(\Omega)$0 is the orthogonal Bergman projection and $A^2(\Omega)$1 is pull-back by the $A^2(\Omega)$2-flow, the $A^2(\Omega)$3th equivariant Bergman kernel is

$A^2(\Omega)$4

and for a fixed closed interval $A^2(\Omega)$5, the partial Bergman kernel is

$A^2(\Omega)$6

This is the kernel of the orthogonal projection onto $A^2(\Omega)$7 (Ioos, 25 Nov 2025).

The paper computes the full off-diagonal asymptotics of these kernels. Near a point $A^2(\Omega)$8, with normal coordinates $A^2(\Omega)$9 and model kernel

$\Omega$00

one has, as $\Omega$01 and uniformly for $\Omega$02 and $\Omega$03 bounded,

$\Omega$04

where $\Omega$05 is the standard Kähler potential. In adapted interface coordinates $\Omega$06, where $\Omega$07 measures the normal direction to $\Omega$08 and $\Omega$09 runs in the horizontal tangent, the partial kernel satisfies

$\Omega$10

and in the deep bulk, with $\Omega$11 sufficiently negative, the error becomes exponentially small and $\Omega$12.

Choosing an orthonormal basis $\Omega$13 of the spectral subspace $\Omega$14 produces an induced $\Omega$15 point process on $\Omega$16 with joint density

$\Omega$17

For a test function $\Omega$18, the linear statistic

$\Omega$19

has exact finite-$\Omega$20 formulas

$\Omega$21

$\Omega$22

As $\Omega$23,

$\Omega$24

where $\Omega$25 is the droplet, and the centered, properly normalized statistic converges in distribution to $\Omega$26.

The variance has the asymptotic expansion

$\Omega$27

which is equivalently described as the sum of an $\Omega$28-norm squared in the bulk and an $\Omega$29-norm squared on the boundary. In the interpretation given there, the leading $\Omega$30 law of large numbers shows that eigenvector zeros become uniformly distributed in the classical droplet $\Omega$31, while the next-order central-limit theorem links the fluctuations to intrinsic Sobolev norms; these are described as spectral-geometric invariants of the quantization datum. The assumptions that $\Omega$32 has bounded geometry at infinity and that $\Omega$33 is regular near $\Omega$34 control the kernel decay and the smoothness of the droplet boundary $\Omega$35 (Ioos, 25 Nov 2025).

Together, these two strands show that “statistical Bergman geometry” names a program in which Bergman kernels organize both information-geometric structures on bounded domains and random point-process statistics in Kähler quantization. In the first setting, the key object is the Fisher pullback along the statistical embedding; in the second, it is the determinantal process induced by partial Bergman kernels. In both, Bergman geometry is read through statistical observables such as expectations, covariances, limit laws, and fluctuation functionals.