Weighted Bergman Metrics

• Weighted Bergman metrics are Kähler metrics defined via weighted Bergman kernels over holomorphic function spaces, extending classical Bergman theory with variable weights.
• This framework facilitates explicit analyses of curvature, biholomorphic invariance, and large-parameter asymptotics that bridge Kähler–Einstein and Fock–Bargmann–Hartogs domain theories.
• Applications include computable models on unit balls, precise boundary behavior, and quotient constructions under symmetry groups, offering insights into both global and local complex geometry.

Weighted Bergman metrics are Kähler metrics derived from weighted Bergman kernels of Hilbert spaces of square-integrable holomorphic functions on complex domains. For a domain $\Omega \subset \mathbb{C}^n$ and an admissible positive weight $\mu$, one considers the weighted Bergman space

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

its reproducing kernel $K_{\Omega,\mu}$, and the metric

$$g_{\Omega,\mu}(z)=\sum_{i,j=1}^n \frac{\partial^2}{\partial z_i \partial \bar z_j}\log K_{\Omega,\mu}(z,z)\,dz_i\otimes d\bar z_j.$$

This framework generalizes the classical Bergman metric by allowing the geometry of the domain, auxiliary potentials, or dynamical kernel constructions to enter through the weight. Current work centers on biholomorphic invariance, extremal descriptions via minimum integrals, large-parameter asymptotics, and explicit constructions on model and quotient domains (Yoo, 2024, Yoo, 17 May 2026, Ghosh, 2021, Sykora, 12 Mar 2025).

1. Analytic definition and basic geometric structure

Let $\Omega \subset \mathbb{C}^n$ be a domain and let $d\lambda$ denote Lebesgue measure. A weight is a positive measurable function $\mu:\Omega\to(0,\infty)$, and the associated weighted measure is $d\mu(z)=\mu(z)\,d\lambda(z)$. The weighted $L^2$ space is

$\mu$0

with inner product

$\mu$1

The weighted Bergman space is then $\mu$2. A standard admissibility condition is that $\mu$3 be closed in $\mu$4 and that point evaluations be continuous; in practice, if $\mu$5 is locally integrable, for instance when $\mu$6 is continuous and positive, then $\mu$7 is admissible. Equivalent bounded-evaluation formulations are also used in the weighted kernel literature (Yoo, 2024, Yoo, 17 May 2026, Jain et al., 2022).

For an admissible weight, the weighted Bergman kernel $\mu$8 is holomorphic in $\mu$9 and anti-holomorphic in $$A^2(\Omega,\mu)=\mathcal O(\Omega)\cap L^2(\Omega,\mu),$$0 and satisfies the reproducing identity

$$A^2(\Omega,\mu)=\mathcal O(\Omega)\cap L^2(\Omega,\mu),$$1

If $$A^2(\Omega,\mu)=\mathcal O(\Omega)\cap L^2(\Omega,\mu),$$2 is an orthonormal basis of $$A^2(\Omega,\mu)=\mathcal O(\Omega)\cap L^2(\Omega,\mu),$$3, then

$$A^2(\Omega,\mu)=\mathcal O(\Omega)\cap L^2(\Omega,\mu),$$4

On the diagonal, the weighted Bergman metric is defined wherever $$A^2(\Omega,\mu)=\mathcal O(\Omega)\cap L^2(\Omega,\mu),$$5 by the potential $$A^2(\Omega,\mu)=\mathcal O(\Omega)\cap L^2(\Omega,\mu),$$6. If this Hermitian form is positive definite everywhere, it is a Kähler metric. For $$A^2(\Omega,\mu)=\mathcal O(\Omega)\cap L^2(\Omega,\mu),$$7,

$$A^2(\Omega,\mu)=\mathcal O(\Omega)\cap L^2(\Omega,\mu),$$8

The associated holomorphic sectional curvature is

$$A^2(\Omega,\mu)=\mathcal O(\Omega)\cap L^2(\Omega,\mu),$$9

with $K_{\Omega,\mu}$0 the curvature tensor of $K_{\Omega,\mu}$1 (Yoo, 2024).

The same formalism extends to more specialized settings. On bounded domains one often writes the measure as $K_{\Omega,\mu}$2 for a weight potential $K_{\Omega,\mu}$3, giving the metric

$K_{\Omega,\mu}$4

and one studies not only sectional curvature but also the holomorphic bisectional curvature

$K_{\Omega,\mu}$5

On product domains with product weights, the kernel factors and the potential splits as a sum, so the metric is the orthogonal sum of the factor metrics (Yoo, 17 May 2026, Jain et al., 2022).

2. Biholomorphic covariance and invariant weight assignments

Weighted Bergman metrics are not automatically invariant under biholomorphisms; the decisive issue is the transformation law of the weight. If $K_{\Omega,\mu}$6 is biholomorphic and the target weight $K_{\Omega,\mu}$7 satisfies

$K_{\Omega,\mu}$8

for some holomorphic nowhere-vanishing $K_{\Omega,\mu}$9 on $$g_{\Omega,\mu}(z)=\sum_{i,j=1}^n \frac{\partial^2}{\partial z_i \partial \bar z_j}\log K_{\Omega,\mu}(z,z)\,dz_i\otimes d\bar z_j.$$0, then the weighted kernel transforms by

$$g_{\Omega,\mu}(z)=\sum_{i,j=1}^n \frac{\partial^2}{\partial z_i \partial \bar z_j}\log K_{\Omega,\mu}(z,z)\,dz_i\otimes d\bar z_j.$$1

where $$g_{\Omega,\mu}(z)=\sum_{i,j=1}^n \frac{\partial^2}{\partial z_i \partial \bar z_j}\log K_{\Omega,\mu}(z,z)\,dz_i\otimes d\bar z_j.$$2. Under the same assumptions, and provided both weighted Bergman metrics are positive definite, one has

$$g_{\Omega,\mu}(z)=\sum_{i,j=1}^n \frac{\partial^2}{\partial z_i \partial \bar z_j}\log K_{\Omega,\mu}(z,z)\,dz_i\otimes d\bar z_j.$$3

Thus weighted Bergman geometry becomes biholomorphically natural only after the weight is organized functorially (Yoo, 2024).

This leads to the notion of an invariant weight assignment on a class of domains $$g_{\Omega,\mu}(z)=\sum_{i,j=1}^n \frac{\partial^2}{\partial z_i \partial \bar z_j}\log K_{\Omega,\mu}(z,z)\,dz_i\otimes d\bar z_j.$$4: an assignment $$g_{\Omega,\mu}(z)=\sum_{i,j=1}^n \frac{\partial^2}{\partial z_i \partial \bar z_j}\log K_{\Omega,\mu}(z,z)\,dz_i\otimes d\bar z_j.$$5 is invariant if for every biholomorphism $$g_{\Omega,\mu}(z)=\sum_{i,j=1}^n \frac{\partial^2}{\partial z_i \partial \bar z_j}\log K_{\Omega,\mu}(z,z)\,dz_i\otimes d\bar z_j.$$6,

$$g_{\Omega,\mu}(z)=\sum_{i,j=1}^n \frac{\partial^2}{\partial z_i \partial \bar z_j}\log K_{\Omega,\mu}(z,z)\,dz_i\otimes d\bar z_j.$$7

for a holomorphic nowhere-vanishing $$g_{\Omega,\mu}(z)=\sum_{i,j=1}^n \frac{\partial^2}{\partial z_i \partial \bar z_j}\log K_{\Omega,\mu}(z,z)\,dz_i\otimes d\bar z_j.$$8. A stronger condition is the canonical assignment of level $$g_{\Omega,\mu}(z)=\sum_{i,j=1}^n \frac{\partial^2}{\partial z_i \partial \bar z_j}\log K_{\Omega,\mu}(z,z)\,dz_i\otimes d\bar z_j.$$9,

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

In that case the normalized density $\Omega \subset \mathbb{C}^n$1 is biholomorphically invariant. This framework was introduced to characterize when weighted Bergman metrics on domains are genuinely biholomorphic invariants (Yoo, 2024).

Two canonical examples organize much of the recent theory. The first is Tian’s Kähler–Einstein assignment on bounded pseudoconvex domains: if $\Omega \subset \mathbb{C}^n$2 is the unique complete Kähler–Einstein metric with Ricci tensor $\Omega \subset \mathbb{C}^n$3, then

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

Because $\Omega \subset \mathbb{C}^n$5 transforms by $\Omega \subset \mathbb{C}^n$6, this yields a canonical assignment of level $\Omega \subset \mathbb{C}^n$7. The second is Tsuji’s dynamical assignment on bounded domains, defined recursively by

$\Omega \subset \mathbb{C}^n$8

and satisfying

$\Omega \subset \mathbb{C}^n$9

Both produce biholomorphically invariant weighted Bergman metrics (Yoo, 2024).

A closely related invariant family appears when the weight is chosen as a negative power of the ordinary Bergman kernel. For $d\lambda$0 on a domain $d\lambda$1, the corresponding kernel transforms as

$d\lambda$2

under a biholomorphism $d\lambda$3, and consequently the weighted Bergman metric is invariant:

$d\lambda$4

This furnishes an explicit family of invariant weighted Bergman metrics in all dimensions, in contrast with general weights, for which invariance typically fails unless the weight is transported appropriately (Jain et al., 2022).

3. Extremal characterizations and curvature identities

A central feature of weighted Bergman geometry is that kernels, metric coefficients, and curvature invariants admit extremal descriptions. For a point $d\lambda$5 and direction $d\lambda$6, the weighted minimum integral method introduces

$d\lambda$7

$d\lambda$8

$d\lambda$9

and the associated minimum integrals

$\mu:\Omega\to(0,\infty)$0

$\mu:\Omega\to(0,\infty)$1

$\mu:\Omega\to(0,\infty)$2

The weighted Bergman–Fuks identities then read

$\mu:\Omega\to(0,\infty)$3

$\mu:\Omega\to(0,\infty)$4

$\mu:\Omega\to(0,\infty)$5

These identities are the basic mechanism behind asymptotic estimates and biholomorphic comparison results (Yoo, 2024).

The bisectional theory refines this extremal structure. For a bounded domain with weight potential $\mu:\Omega\to(0,\infty)$6, one introduces minimum integrals

$\mu:\Omega\to(0,\infty)$7

defined by imposing mixed first- and second-order vanishing conditions. The key curvature identity is

$\mu:\Omega\to(0,\infty)$8

where

$\mu:\Omega\to(0,\infty)$9

Moreover,

$d\mu(z)=\mu(z)\,d\lambda(z)$0

and $d\mu(z)=\mu(z)\,d\lambda(z)$1 (Yoo, 17 May 2026).

An equivalent Hilbert-space description uses $d\mu(z)=\mu(z)\,d\lambda(z)$2-orthogonal projections. Evaluation and derivative functionals are represented by derivatives of the reproducing kernel, and the reciprocal minimum integrals become squared norms of projected representers. This identifies the extremizers as orthogonal projections of kernel derivatives onto subspaces determined by vanishing constraints. A plausible implication is that the geometry of weighted Bergman metrics is especially amenable to quantitative comparison, because both curvature and metric lengths can be reduced to operator-theoretic norms rather than pointwise kernel differentiation alone (Yoo, 17 May 2026).

4. Asymptotic expansions and convergence regimes

A major recent development is a domain version of the Tian–Yau–Zelditch expansion for weighted Bergman kernels and metrics. Let $d\mu(z)=\mu(z)\,d\lambda(z)$3 be pseudoconvex, let $d\mu(z)=\mu(z)\,d\lambda(z)$4 be smooth strictly plurisubharmonic, and define $d\mu(z)=\mu(z)\,d\lambda(z)$5. For weights of the form $d\mu(z)=\mu(z)\,d\lambda(z)$6, assuming the weighted Bergman metric for $d\mu(z)=\mu(z)\,d\lambda(z)$7 is positive definite, the weighted kernel, metric, and holomorphic sectional curvature admit large-$d\mu(z)=\mu(z)\,d\lambda(z)$8 expansions. In particular,

$d\mu(z)=\mu(z)\,d\lambda(z)$9

and

$L^2$0

For $L^2$1 one obtains corresponding adjusted expansions, and in all cases the consequences include

$L^2$2

as $L^2$3 (Yoo, 2024).

The natural global setting for uniform convergence is the class of uniform squeezing domains. A bounded domain $L^2$4 is uniform squeezing if there exists $L^2$5 such that for every $L^2$6 there is a biholomorphism $L^2$7 with $L^2$8 and

$L^2$9

Uniform squeezing implies pseudoconvexity and existence of a unique complete Kähler–Einstein metric on $\mu$00; by Yeung’s theorem, $\mu$01 has bounded geometry of infinite order, and $\mu$02 has uniformly bounded $\mu$03 norms in normalized coordinates. This bounded geometry is the input that upgrades pointwise TYZ-type asymptotics to uniform estimates (Yoo, 2024).

For Tian’s sequence, defined from the Kähler–Einstein weights

$\mu$04

the normalized quantities

$\mu$05

converge uniformly on uniform squeezing domains to $\mu$06, $\mu$07, and $\mu$08, respectively. For Tsuji’s modified dynamical sequence, with

$\mu$09

the paper proves uniform convergence at the potential level:

$\mu$10

with $\mu$11 at rate $\mu$12. Metric convergence for this modified Tsuji sequence is not established there (Yoo, 2024).

A related semiclassical expansion studies weights of the form

$\mu$13

where $\mu$14 for the Kähler metric $\mu$15, and $\mu$16 are real-analytic. In that setting,

$\mu$17

and the weighted Bergman metric satisfies

$\mu$18

This gives an explicit first-order description of how the auxiliary weight $\mu$19 and scalar curvature enter the weighted metric (Sykora, 12 Mar 2025).

5. Explicit models and computable families

The unit ball provides the most explicit model. On $\mu$20 with weight

$\mu$21

the weighted Bergman kernel is

$\mu$22

where

$\mu$23

The corresponding weighted Bergman metric is

$\mu$24

and its holomorphic sectional curvature is constant:

$\mu$25

This model isolates the effect of the weight parameter on scale and curvature (Yoo, 2024).

On the unit disc, the family $\mu$26 yields exact closed forms. Since

$\mu$27

one has

$\mu$28

hence

$\mu$29

The weighted Bergman metric is

$\mu$30

so

$\mu$31

Its Gaussian curvature is $\mu$32, and completeness is preserved because the metric is a positive constant multiple of the complete Bergman metric (Jain et al., 2022).

Other explicit weighted metrics arise from tube domains and Hartogs-type domains. For the unit ball or higher-dimensional ball with radial weight $\mu$33, the metric is a constant multiple of the unweighted Bergman metric, with holomorphic sectional curvature $\mu$34 in the normalization used there. Analogous formulas hold on the Siegel domain $\mu$35 and on tube domains over the Lorentz cone, where the diagonal kernel has the form $\mu$36 and therefore

$\mu$37

These examples make curvature scaling and completeness immediate (Deng et al., 2020).

The Fock–Bargmann–Hartogs domains

$\mu$38

supply an unbounded, nonhomogeneous setting with an explicit weighted kernel. For the Kähler potential

$\mu$39

and the weighted Hilbert space with weight $\mu$40, the weighted Bergman kernel is computed explicitly for $\mu$41. The associated Rawnsley $\mu$42-function is constant, and thus $\mu$43 is balanced, if and only if

$\mu$44

In that case

$\mu$45

for some constant $\mu$46, so the weighted Bergman metric coincides exactly with the scaled Kähler metric:

$\mu$47

This identifies a class of weighted Bergman metrics that are simultaneously balanced and projectively induced (Bi et al., 2015).

A common source of confusion concerns curvature normalization on the ball. In the classical convention, the Bergman metric on $\mu$48 has constant holomorphic sectional curvature $\mu$49, whereas in the KE-weighted setting of the bisectional-curvature analysis one obtains

$\mu$50

For $\mu$51 this is $\mu$52. The discrepancy reflects differing global normalization choices; in the latter case the normalization is adapted to the weights tied to $\mu$53 and Ricci $\mu$54 scaling (Yoo, 17 May 2026).

6. Boundary asymptotics, quotient constructions, and open directions

Boundary behavior depends sharply on the weight class. For a bounded planar domain $\mu$55, a $\mu$56-smooth boundary point $\mu$57, and a weight $\mu$58 that extends continuously to $\mu$59 with $\mu$60, one has

$\mu$61

as $\mu$62, where $\mu$63 is the Euclidean distance to $\mu$64. If $\mu$65 is simply connected, then for all integers $\mu$66,

$\mu$67

The same work also proves additive and multiplicative boundary rules for weighted kernels as functions of the weight, and for the family $\mu$68 establishes

$\mu$69

near smooth boundary points (Jain et al., 2022).

For bounded pseudoconvex domains equipped with the KE-weighted Bergman metrics, the squeezing function controls curvature quantitatively. If $\mu$70 is the squeezing function and

$\mu$71

then the weighted bisectional curvature $\mu$72 satisfies explicit two-sided inequalities in terms of powers of $\mu$73. At strongly pseudoconvex boundary points, where $\mu$74, one obtains asymptotic coincidence with the unit ball:

$\mu$75

Sending $\mu$76 on uniformly squeezing domains yields the corresponding asymptotic for the Kähler–Einstein metric itself, recovering the known boundary behavior of KE bisectional curvature without Fefferman expansions or Pinchuk-type scaling (Yoo, 17 May 2026).

Weighted Bergman metrics also descend to quotient domains. If a finite pseudoreflection group $\mu$77 acts linearly on a bounded $\mu$78-invariant domain $\mu$79, the Chevalley–Shephard–Todd theorem provides a basic polynomial map

$\mu$80

with $\mu$81 proper and deck group $\mu$82. For weights of the form $\mu$83, the quotient kernel can be reconstructed from the source kernel by representation-weighted group averages. For a one-dimensional representation $\mu$84 with generating polynomial $\mu$85,

$\mu$86

Accordingly,

$\mu$87

where $\mu$88 and $\mu$89 is the corresponding diagonal averaged kernel. In the sign representation, the Jacobian $\mu$90 appears explicitly, so the pullback metric records the orbifold branch structure (Ghosh, 2021).

Several open directions are explicit in the literature. The domain TYZ expansion currently requires smooth strictly plurisubharmonic potentials on pseudoconvex domains and positive definiteness of the weighted Bergman metric; extending uniform convergence beyond uniform squeezing domains would require comparable bounded-geometry hypotheses. For Tsuji’s modified sequence, only potential-level convergence is proved, not convergence of the weighted Bergman metrics themselves. In the boundary theory on planar domains, derivative asymptotics were proved under a simply connected hypothesis, which the authors expect can be removed. For general weights vanishing at the boundary, beyond the specific family $\mu$91, precise asymptotics remain wide open (Yoo, 2024, Jain et al., 2022).

Weighted Bergman metrics therefore occupy a position between several mature theories: classical Bergman geometry, Kähler–Einstein geometry, semiclassical kernel asymptotics, and quotient/orbifold constructions. The modern viewpoint is that the metric is best understood not as a single canonical object attached to a domain, but as a family of biholomorphically meaningful Kähler structures whose invariance, curvature, and asymptotics are governed by the analytic structure of the weight.