Bergman Metric in Complex Geometry

• Bergman metric is a canonical Kähler metric defined from the logarithm of the Bergman kernel, serving as a biholomorphic invariant in complex geometry.
• It underpins projective, diastatic, and statistical realizations, linking complex analysis with information geometry and Kähler immersion theory.
• Explicit studies reveal its variable curvature behavior and rigidity properties, particularly in bounded domains and symmetric spaces.

The Bergman metric is the canonical Kähler metric attached to the Bergman kernel of a complex domain or, more generally, of a complex manifold whose $L^2$-holomorphic theory is sufficiently rich. On a bounded domain $D\subset \mathbb C^n$, if $K_D(z,\bar z)$ denotes the Bergman kernel on the diagonal, the metric is defined by the Kähler potential $\log K_D(z,\bar z)$, so that

$$\omega_D=\frac{i}{2}\partial\bar\partial\log K_D \quad\text{or, in another common normalization,}\quad \omega_D=\sqrt{-1}\,\partial\bar\partial\log K_D,$$

with normalization depending on the source. In coordinates,

$$(g_D)_{i\bar j}=\frac{\partial^2}{\partial z_i\partial \bar z_j}\log K_D(z,\bar z).$$

This metric is biholomorphically invariant and has become a central object in several complex variables, Kähler geometry, uniformization theory, and, more recently, information geometry and projective-immersion theory (Loi et al., 21 Jan 2026, Huang et al., 2016, Cho et al., 2023).

1. Kernel-theoretic definition and general framework

For a bounded domain $\Omega\subset \mathbb C^n$, the Bergman space is

$A^2(\Omega)=\mathcal O(\Omega)\cap L^2(\Omega),$

or, in the top-form formulation used on general complex manifolds,

$$A^2(\Omega)=\left\{\alpha\in A^n(\operatorname{Reg}(\Omega)):\sqrt{-1}^{\,n^2}\int \alpha\wedge \overline\alpha<\infty\right\}.$$

If $\{\phi_\alpha\}$ is a complete orthonormal system, then the Bergman kernel is

$D\subset \mathbb C^n$0

with the reproducing property

$D\subset \mathbb C^n$1

Its diagonal value $D\subset \mathbb C^n$2 is strictly plurisubharmonic when the Bergman metric is defined, and $D\subset \mathbb C^n$3 is then a Kähler potential (Loi et al., 21 Jan 2026, Huang et al., 2016).

On general complex manifolds, the relevant intrinsic hypotheses are that the Bergman kernel be nowhere vanishing and that the Bergman space separate holomorphic directions. In the language used for complex manifolds beyond bounded domains, the Bergman form defines a biholomorphically invariant Kähler metric exactly under base-point freeness and separation of holomorphic directions (Huang et al., 2023).

A basic model is the unit ball $D\subset \mathbb C^n$4, whose Bergman kernel is

$D\subset \mathbb C^n$5

Its Bergman metric is a scalar multiple of the complex hyperbolic metric. Depending on normalization, the holomorphic sectional curvature is written either as $D\subset \mathbb C^n$6 or $D\subset \mathbb C^n$7; the literature cited here explicitly notes this normalization dependence (Loi et al., 21 Jan 2026, Ebenfelt et al., 20 Feb 2025).

2. Projective, diastatic, and statistical realizations

The Bergman metric admits a natural projective realization. Given an orthonormal basis $D\subset \mathbb C^n$8 of $D\subset \mathbb C^n$9, the Bergman–Bochner map

$K_D(z,\bar z)$0

is a full Kähler immersion, and the Bergman metric is the pullback of the Fubini–Study metric. In the same spirit, for complex manifolds with Bergman metric, the Bergman–Bochner map realizes

$K_D(z,\bar z)$1

making projective geometry a basic tool in Bergman-metric rigidity problems (Loi et al., 21 Jan 2026, Huang et al., 2023).

A second canonical object is Calabi’s diastasis. For the Bergman metric it has the explicit form

$K_D(z,\bar z)$2

This formula is central in rigidity arguments based on Calabi’s criterion, finite-dimensional projective inducedness, and Kähler immersions into complex space forms (Palmieri, 20 Oct 2025).

A further development is the Bergman dual. For a bounded domain $K_D(z,\bar z)$3 centered at the origin, one sets

$K_D(z,\bar z)$4

defines $K_D(z,\bar z)$5 to be the maximal domain on which $K_D(z,\bar z)$6 and $K_D(z,\bar z)$7 is defined, and then defines

$K_D(z,\bar z)$8

For the ball,

$K_D(z,\bar z)$9

and $\log K_D(z,\bar z)$0 is the Fubini–Study form on the affine chart of $\log K_D(z,\bar z)$1, up to scaling. This construction was introduced as a generalization of the bounded-symmetric-domain/compact-dual correspondence (Loi et al., 7 Oct 2025).

More recently, the Bergman metric has been recast in information-geometric terms. For a bounded domain $\log K_D(z,\bar z)$2, the embedding

$\log K_D(z,\bar z)$3

maps $\log K_D(z,\bar z)$4 into a space of probability measures on $\log K_D(z,\bar z)$5, and the pullback of the Fisher information metric satisfies

$\log K_D(z,\bar z)$6

In the same framework, Calabi’s diastasis becomes the pullback of Kullback divergence: $\log K_D(z,\bar z)$7 This identifies the Bergman metric as an information metric and the diastasis as a statistical divergence (Cho et al., 2023).

3. Curvature, Einsteinity, and explicit geometric behavior

The Bergman metric is Kähler by construction, but its curvature behavior varies sharply with the domain. On bounded homogeneous domains, the Bergman metric is Kähler–Einstein with Einstein constant $\log K_D(z,\bar z)$8, and this fact is used as a structural input in several Hartogs-domain rigidity theorems (Mossa, 17 Apr 2026).

At the opposite end, constant-curvature phenomena are highly constrained. For bounded domains or complex manifolds with sufficiently rich Bergman space, the negative constant-holomorphic-sectional-curvature case is rigid: the only possible value is that of the ball, and the geometry is forced to be ball-like up to a negligible exceptional set (Ebenfelt et al., 20 Feb 2025, Huang et al., 2023). The zero-curvature case is excluded in the classification of Bergman metrics with constant holomorphic sectional curvature under the standard nondegeneracy assumptions (Huang et al., 2023).

Positive constant curvature is more subtle. For every pair of positive integers $\log K_D(z,\bar z)$9 with $$\omega_D=\frac{i}{2}\partial\bar\partial\log K_D \quad\text{or, in another common normalization,}\quad \omega_D=\sqrt{-1}\,\partial\bar\partial\log K_D,$$0, there exists an $$\omega_D=\frac{i}{2}\partial\bar\partial\log K_D \quad\text{or, in another common normalization,}\quad \omega_D=\sqrt{-1}\,\partial\bar\partial\log K_D,$$1-parameter family of Reinhardt domains in $$\omega_D=\frac{i}{2}\partial\bar\partial\log K_D \quad\text{or, in another common normalization,}\quad \omega_D=\sqrt{-1}\,\partial\bar\partial\log K_D,$$2 whose Bergman metrics are locally isometric to $$\omega_D=\frac{i}{2}\partial\bar\partial\log K_D \quad\text{or, in another common normalization,}\quad \omega_D=\sqrt{-1}\,\partial\bar\partial\log K_D,$$3 times the Fubini–Study metric, with Bergman kernel

$$\omega_D=\frac{i}{2}\partial\bar\partial\log K_D \quad\text{or, in another common normalization,}\quad \omega_D=\sqrt{-1}\,\partial\bar\partial\log K_D,$$4

With the normalization used there, the Fubini–Study metric has constant holomorphic sectional curvature $$\omega_D=\frac{i}{2}\partial\bar\partial\log K_D \quad\text{or, in another common normalization,}\quad \omega_D=\sqrt{-1}\,\partial\bar\partial\log K_D,$$5, so $$\omega_D=\frac{i}{2}\partial\bar\partial\log K_D \quad\text{or, in another common normalization,}\quad \omega_D=\sqrt{-1}\,\partial\bar\partial\log K_D,$$6 has curvature $$\omega_D=\frac{i}{2}\partial\bar\partial\log K_D \quad\text{or, in another common normalization,}\quad \omega_D=\sqrt{-1}\,\partial\bar\partial\log K_D,$$7. These examples show that positive-curvature Bergman metrics are abundant in the noncomplete setting (Bhat et al., 16 May 2026).

Explicit nonhomogeneous examples also reveal finer curvature phenomena. On the symmetrized bidisc

$$\omega_D=\frac{i}{2}\partial\bar\partial\log K_D \quad\text{or, in another common normalization,}\quad \omega_D=\sqrt{-1}\,\partial\bar\partial\log K_D,$$8

the Bergman metric has holomorphic sectional curvature negatively pinched, with global bounds

$$\omega_D=\frac{i}{2}\partial\bar\partial\log K_D \quad\text{or, in another common normalization,}\quad \omega_D=\sqrt{-1}\,\partial\bar\partial\log K_D,$$9

but its holomorphic bisectional curvature is not negatively pinched and is positive at some points. Thus a complete Bergman metric can have uniformly negative holomorphic sectional curvature while still exhibiting positive holomorphic bisectional curvature somewhere (Cho et al., 2020).

In the information-geometric formulation, the holomorphic sectional curvature admits a statistical expression. In holomorphic normal coordinates,

$$(g_D)_{i\bar j}=\frac{\partial^2}{\partial z_i\partial \bar z_j}\log K_D(z,\bar z).$$0

which immediately yields the upper bound $$(g_D)_{i\bar j}=\frac{\partial^2}{\partial z_i\partial \bar z_j}\log K_D(z,\bar z).$$1 in that normalization (Cho et al., 2023).

4. Rigidity and uniformization theorems

One of the main themes in the modern theory is that special curvature conditions on the Bergman metric force strong geometric uniformization.

For smoothly bounded strongly pseudoconvex domains in $$(g_D)_{i\bar j}=\frac{\partial^2}{\partial z_i\partial \bar z_j}\log K_D(z,\bar z).$$2, Cheng’s conjecture is solved affirmatively: the Bergman metric is Kähler–Einstein if and only if the domain is biholomorphic to the unit ball (Huang et al., 2016). In dimension $$(g_D)_{i\bar j}=\frac{\partial^2}{\partial z_i\partial \bar z_j}\log K_D(z,\bar z).$$3, an algebraic version for Stein spaces with isolated normal singularities states that the Bergman metric on the regular part is Kähler–Einstein if and only if the space is biholomorphic to $$(g_D)_{i\bar j}=\frac{\partial^2}{\partial z_i\partial \bar z_j}\log K_D(z,\bar z).$$4; equivalently, for finite fixed-point-free ball quotients $$(g_D)_{i\bar j}=\frac{\partial^2}{\partial z_i\partial \bar z_j}\log K_D(z,\bar z).$$5, the Bergman metric is Kähler–Einstein only when $$(g_D)_{i\bar j}=\frac{\partial^2}{\partial z_i\partial \bar z_j}\log K_D(z,\bar z).$$6 is trivial (Ganguly et al., 2022).

Constant negative holomorphic sectional curvature leads to a sharp uniformization theorem. If a bounded domain $$(g_D)_{i\bar j}=\frac{\partial^2}{\partial z_i\partial \bar z_j}\log K_D(z,\bar z).$$7 has Bergman metric of constant holomorphic sectional curvature $$(g_D)_{i\bar j}=\frac{\partial^2}{\partial z_i\partial \bar z_j}\log K_D(z,\bar z).$$8, then $$(g_D)_{i\bar j}=\frac{\partial^2}{\partial z_i\partial \bar z_j}\log K_D(z,\bar z).$$9 is biholomorphic to a domain $\Omega\subset \mathbb C^n$0 such that $\Omega\subset \mathbb C^n$1 is relatively closed of measure zero, every $\Omega\subset \mathbb C^n$2-holomorphic function on $\Omega\subset \mathbb C^n$3 extends to one on $\Omega\subset \mathbb C^n$4, and

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

Moreover,

$\Omega\subset \mathbb C^n$6

This generalizes Lu’s classical complete-case theorem (Ebenfelt et al., 20 Feb 2025).

Local symmetry yields a higher-rank analogue. If $\Omega\subset \mathbb C^n$7 is bounded and the Bergman metric is locally symmetric, meaning $\Omega\subset \mathbb C^n$8, then completeness forces $\Omega\subset \mathbb C^n$9 to be a bounded symmetric domain. If one assumes only pseudoconvexity, then

$A^2(\Omega)=\mathcal O(\Omega)\cap L^2(\Omega),$0

where $A^2(\Omega)=\mathcal O(\Omega)\cap L^2(\Omega),$1 is a bounded symmetric domain and $A^2(\Omega)=\mathcal O(\Omega)\cap L^2(\Omega),$2 is relatively closed and pluripolar (Loi et al., 21 Jan 2026).

The same rigidity persists in explicit Hartogs families. For Hartogs domains over bounded homogeneous bases,

$A^2(\Omega)=\mathcal O(\Omega)\cap L^2(\Omega),$3

if $A^2(\Omega)=\mathcal O(\Omega)\cap L^2(\Omega),$4 and the Bergman metric is Kähler–Einstein, then $A^2(\Omega)=\mathcal O(\Omega)\cap L^2(\Omega),$5 is biholomorphic to the unit ball $A^2(\Omega)=\mathcal O(\Omega)\cap L^2(\Omega),$6 (Mossa, 17 Apr 2026). For Cartan–Hartogs domains,

$A^2(\Omega)=\mathcal O(\Omega)\cap L^2(\Omega),$7

the following are equivalent: $A^2(\Omega)=\mathcal O(\Omega)\cap L^2(\Omega),$8; the Bergman metric is Kähler–Einstein; the Bergman metric is a Kähler–Ricci soliton, for $A^2(\Omega)=\mathcal O(\Omega)\cap L^2(\Omega),$9; and some positive multiple of the Bergman dual metric is finitely projectively induced (Loi et al., 7 Oct 2025).

5. Hartogs constructions, canonical comparisons, and boundary obstructions

A major strength of Bergman-metric theory is the availability of explicit kernel formulas on structured nonhomogeneous domains. For Cartan–Hartogs domains, the Bergman kernel can be written in terms of a one-variable function

$$A^2(\Omega)=\left\{\alpha\in A^n(\operatorname{Reg}(\Omega)):\sqrt{-1}^{\,n^2}\int \alpha\wedge \overline\alpha<\infty\right\}.$$0

and even in rational form

$$A^2(\Omega)=\left\{\alpha\in A^n(\operatorname{Reg}(\Omega)):\sqrt{-1}^{\,n^2}\int \alpha\wedge \overline\alpha<\infty\right\}.$$1

This reduction makes the Einstein equation and projective-inducedness questions algebraically tractable (Loi et al., 7 Oct 2025).

The Bergman metric can behave very differently from other natural canonical metrics on the same Hartogs domain. On $$A^2(\Omega)=\left\{\alpha\in A^n(\operatorname{Reg}(\Omega)):\sqrt{-1}^{\,n^2}\int \alpha\wedge \overline\alpha<\infty\right\}.$$2, the metric

$$A^2(\Omega)=\left\{\alpha\in A^n(\operatorname{Reg}(\Omega)):\sqrt{-1}^{\,n^2}\int \alpha\wedge \overline\alpha<\infty\right\}.$$3

is Kähler–Einstein exactly when

$$A^2(\Omega)=\left\{\alpha\in A^n(\operatorname{Reg}(\Omega)):\sqrt{-1}^{\,n^2}\int \alpha\wedge \overline\alpha<\infty\right\}.$$4

whereas the Bergman metric is Kähler–Einstein only in the ball case. A second metric $$A^2(\Omega)=\left\{\alpha\in A^n(\operatorname{Reg}(\Omega)):\sqrt{-1}^{\,n^2}\int \alpha\wedge \overline\alpha<\infty\right\}.$$5 has projectively induced behavior for $$A^2(\Omega)=\left\{\alpha\in A^n(\operatorname{Reg}(\Omega)):\sqrt{-1}^{\,n^2}\int \alpha\wedge \overline\alpha<\infty\right\}.$$6 and sufficiently large scaling, again in sharp contrast with the Bergman metric (Loi et al., 7 Oct 2025).

There is also a distinct rigidity problem for Bergman metrics induced by finite-dimensional balls. If $$A^2(\Omega)=\left\{\alpha\in A^n(\operatorname{Reg}(\Omega)):\sqrt{-1}^{\,n^2}\int \alpha\wedge \overline\alpha<\infty\right\}.$$7 admits a holomorphic isometric immersion into $$A^2(\Omega)=\left\{\alpha\in A^n(\operatorname{Reg}(\Omega)):\sqrt{-1}^{\,n^2}\int \alpha\wedge \overline\alpha<\infty\right\}.$$8, then in several broad families the only possibility is again the ball. This is proved for strictly pseudoconvex domains in $$A^2(\Omega)=\left\{\alpha\in A^n(\operatorname{Reg}(\Omega)):\sqrt{-1}^{\,n^2}\int \alpha\wedge \overline\alpha<\infty\right\}.$$9 under a smooth transversality hypothesis, and also for Hartogs domains over bounded homogeneous bases and egg domains over irreducible symmetric bases (Palmieri, 20 Oct 2025).

Boundary singularities can obstruct Einsteinity even more forcefully. If a pseudoconvex domain in $\{\phi_\alpha\}$0, $\{\phi_\alpha\}$1, has a strongly pseudoconvex polyhedral boundary point, then its Bergman metric is well-defined on a nonempty open set but cannot be Einstein on any open subset. The argument identifies the Einstein condition with constancy of the Bergman invariant

$\{\phi_\alpha\}$2

then shows that anisotropic blow-up at such a corner converges to a product model $\{\phi_\alpha\}$3, whose invariant differs from the ball value forced by Einsteinity (Huang et al., 9 Dec 2025).

A different comparison principle appears on complete noncompact Kähler manifolds with complete Bergman metric of bounded curvature. If the ratio

$\{\phi_\alpha\}$4

is bounded on a fundamental domain, then the Bergman metric is uniformly equivalent to a complete Kähler–Einstein metric of negative scalar curvature. This method applies, for example, to the explicit domains

$\{\phi_\alpha\}$5

where boundary limits of Bergman holomorphic sectional curvature are not well-defined and older curvature-limit methods fail (Cho et al., 2021).

6. Related constructions, completeness theory, and broader extensions

The Bergman metric has inspired several closely related structures. One is the curvature-modified metric

$\{\phi_\alpha\}$6

where $\{\phi_\alpha\}$7 is the Bergman metric tensor. This metric is positive definite because the Ricci curvature of the Bergman metric is bounded above by $\{\phi_\alpha\}$8, and it admits its own Kobayashi-type completeness criterion. In particular, bounded hyperconvex domains are complete with respect to $\{\phi_\alpha\}$9, paralleling classical Bergman completeness theory (Dinew, 2012).

Another comparative direction concerns bounded symmetric domains and invariant metrics other than the Bergman metric. A projectively defined Hilbert-type metric on the four classical Cartan families has infinitesimal norm of $D\subset \mathbb C^n$00-type in the singular values, whereas the Bergman metric is $D\subset \mathbb C^n$01-type. Except in rank one, the two metrics differ, even up to scale (Falbel et al., 2024).

There are also nonclassical extensions of the Bergman-metric philosophy outside complex analysis in the usual sense. A Riemannian analogue replaces holomorphic sections by low-frequency Laplace eigenfunctions of a fixed reference metric, producing “Riemannian Bergman metrics of degree $D\subset \mathbb C^n$02” as pullbacks of Euclidean metrics under eigenfunction embeddings. These finite-dimensional symmetric-space approximations are designed as analogues of Bergman approximations in Kähler geometry (Potash, 2013). This suggests a broader “Bergman” paradigm in which canonical metrics arise from kernel or spectral projectors rather than from curvature ansätze alone.

The current literature therefore presents the Bergman metric as more than a canonical Kähler metric attached to $D\subset \mathbb C^n$03-holomorphic data. It is a biholomorphic invariant, a projectively induced metric, a carrier of diastatic and statistical structures, a sensitive detector of boundary and singularity geometry, and a source of rigidity results that characterize balls and bounded symmetric domains across a wide range of complex-analytic settings (Loi et al., 21 Jan 2026, Cho et al., 2023).