Bergman kernel and Bergman metric

Intuition Beginner

Imagine all the holomorphic functions on a domain that are not too large — those whose squared size, integrated over the whole domain, comes out finite. This collection forms a space with a notion of length and angle, much like ordinary vectors in the plane have length and angle. The remarkable fact is that inside this space there lives one special object, built once for the domain, that lets you recover the value of any function at any point by a single integration.

That special object is the Bergman kernel. Pick a point inside the domain. The kernel hands you a particular function whose inner product with anything in the space reads off the value at your chosen point. So the kernel is a universal value-reader: feed it a point, integrate, and it returns the function's value there. Because it is assembled only from the geometry of the domain, it carries deep information about the shape of that domain.

The second gift is a way to measure distance. The kernel changes from point to point, and the rate at which it changes defines a ruler — the Bergman metric — that bends and stretches to fit the domain. Two domains that look the same to holomorphic maps wear the same ruler.

Visual Beginner

A bounded domain in two complex dimensions, drawn as a smooth blob. At an interior point sits a small spike representing the kernel's value-reading function peaked near that point; an arrow labelled "integrate against any holomorphic function" connects the spike to the function's value at the point. Near the boundary the spike grows sharply taller, a cartoon of the kernel blowing up as the point approaches the edge. A second overlay shows tiny ellipses — the unit balls of the Bergman ruler — shrinking and elongating as they near the boundary.

Worked example Beginner

Take the simplest interesting domain: the unit disc in one complex variable, the set of points with size less than one. The holomorphic functions there with finite integrated square are exactly the power series whose coefficients shrink fast enough. A short computation with these power series — integrating each monomial against itself over the disc — produces a single closed formula for the kernel: the value-reading function at a point $w$ is $1$ divided by $\pi$ times $(1-z\bar{w})$ squared.

Check the most basic case. Put the chosen point at the centre, where $w=0$. The formula collapses to $1/\pi$, a constant. Integrating a holomorphic function against this constant over the disc returns the function's average, which for holomorphic functions equals the value at the centre. The mean-value property you already know is the kernel doing its job at one point.

Now slide the point toward the boundary, so $z\bar{w}$ approaches $1$. The denominator shrinks toward zero and the kernel's size grows without bound. This blow-up is universal: every bounded domain's kernel grows as the point nears the edge.

What this tells us: the kernel is a concrete, computable function, equal to a clean formula on the disc, and its boundary blow-up already encodes the shape of the domain near its edge.

Check your understanding Beginner

Formal definition Intermediate+

Let $\Omega \subseteq {\mathbb{C}}^n$ be a domain (open, connected). The Bergman space is

$$A^2(\Omega )=\mathcal{O}(\Omega )\cap L^2(\Omega )=\left\{f\in \mathcal{O}(\Omega ):\Vert f{\Vert }^2={\int }_{\Omega }|f(z){|}^{2}dV(z)<\infty \right\},$$

where $dV$ is Lebesgue measure on ${\mathbb{C}}^n\cong {\mathbb{R}}^{2n}$ and $\mathcal{O}(\Omega )$ denotes the holomorphic functions on $\Omega$. With the inner product $⟨f,g⟩={\int }_{\Omega }f\bar{g}dV$, the space $A^2(\Omega )$ is a separable complex Hilbert space: it is a closed subspace of $L^2(\Omega )$, because the sub-mean-value inequality for $|f{|}^2$ over a polydisc or ball $\overline{B(z,r)}\subset \Omega$ gives the interior estimate

$$|f(z)|\le \frac{1}{\sqrt{V(B(z,r))}}\Vert f{\Vert }_{L^2(B(z,r))}\le \frac{1}{\sqrt{V(B(z,r))}}\Vert f\Vert ,$$

so $L^2$-convergence in $A^2(\Omega )$ forces locally uniform convergence, and the limit stays holomorphic by the Weierstrass convergence theorem in several variables 06.07.01.

The estimate shows that for each fixed $z\in \Omega$ the evaluation functional ${\mathrm{ev}}_z:f\mapsto f(z)$ is bounded on $A^2(\Omega )$. By the Riesz representation theorem there is a unique element $K_{\Omega }(\cdot ,z)\in A^2(\Omega )$ with

$$f(z)=⟨f,K_{\Omega }(\cdot ,z)⟩={\int }_{\Omega }f(\zeta )\overline{K_{\Omega }(\zeta ,z)}dV(\zeta )\text{for all }f\in A^2(\Omega ).$$

The function $K_{\Omega }:\Omega \times \Omega \to \mathbb{C}$, $K_{\Omega }(z,w)=K_{\Omega }(z,\cdot )$ evaluated at $w$ in the natural normalisation, is the Bergman kernel of $\Omega$. It is holomorphic in $z$, antiholomorphic in $w$, conjugate-symmetric $K_{\Omega }(z,w)=\overline{K_{\Omega }(w,z)}$, and positive on the diagonal: $K_{\Omega }(z,z)=\Vert K_{\Omega }(\cdot ,z){\Vert }^2>0$ whenever $A^2(\Omega )\ne \{0\}$, which holds for every bounded $\Omega$ since constants lie in $A^2(\Omega )$.

When $\Omega$ is bounded (so $K_{\Omega }(z,z)>0$ everywhere), the Bergman metric is the Hermitian form with coefficients

$$g_{j\bar{k}}(z)=\frac{{\partial }^2}{\partial z_j\partial {\bar{z}}_k}\log K_{\Omega }(z,z),d{s}^2=\sum _{j,k=1}^n{g}_{j\bar{k}}(z)d{z}_{j}d{\bar{z}}_k.$$

The matrix $(g_{j\bar{k}})$ is positive definite when $\Omega$ is bounded, so $d{s}^2$ is a genuine Kähler metric (its associated $(1,1)$-form is closed, being $\frac{i}{2}\partial \bar{\partial }\log K$).

Counterexamples to common slips

• $A^2(\Omega )$ can be $\{0\}$: for $\Omega ={\mathbb{C}}^n$ the only $L^2$ entire function is $0$ (a non-zero entire function fails to decay), so the kernel and the metric are undefined. Boundedness, or at least $A^2\ne \{0\}$, is essential.
• $K_{\Omega }(z,z)$ is not the integral of $K_{\Omega }(z,w)$ over $w$; the diagonal value is a squared norm, not a total mass. The reproducing identity pairs $f$ against $\overline{K(\cdot ,z)}$, not against $K(z,z)$.
• The metric uses $\log K(z,z)$, not $K(z,z)$. Dropping the logarithm destroys the biholomorphic invariance, because the Jacobian factors in the transformation law cancel only after taking the logarithm and a mixed second derivative.

Key theorem with proof Intermediate+

Theorem (Bergman 1933, transformation law and invariance of the metric). Let $\Phi :{\Omega }_1\to {\Omega }_2$ be a biholomorphic map of domains in ${\mathbb{C}}^n$, with complex Jacobian determinant $J_{\Phi }(z)=\det (\partial {\Phi }_i/\partial z_j)$. Then the Bergman kernels transform by

$$K_{{\Omega }_1}(z,w)=J_{\Phi }(z)K_{{\Omega }_2}(\Phi (z),\Phi (w))\overline{J_{\Phi }(w)},$$

and consequently the Bergman metric is an isometry invariant: $\Phi^ (ds^2_{\Omega_2}) = ds^2_{\Omega_1}$.*

Proof. The change-of-variables formula for a biholomorphism $\Phi$ reads $dV(\Phi (z))=|J_{\Phi }(z){|}^{2}dV(z)$, because the real Jacobian of a holomorphic map equals the squared modulus of the complex Jacobian determinant. Hence the map

$$U:A^2({\Omega }_2)\to A^2({\Omega }_1),(Uf)(z)=J_{\Phi }(z)\cdot f(\Phi (z)),$$

is a surjective linear isometry: for $f\in A^2({\Omega }_2)$,

$$\Vert Uf{\Vert }_{{\Omega }_1}^2={\int }_{{\Omega }_1}|J_{\Phi }(z){|}^2|f(\Phi (z)){|}^{2}dV(z)={\int }_{{\Omega }_2}|f(w){|}^{2}dV(w)=\Vert f{\Vert }_{{\Omega }_2}^2,$$

with inverse $U^{-1}g=(J_{{\Phi }^{-1}}\cdot g)\circ {\Phi }^{-1}$. A unitary intertwiner sends reproducing kernel to reproducing kernel: writing $k_z^{(i)}=K_{{\Omega }_i}(\cdot ,z)$, the defining property $⟨f,k_z^{(1)}{⟩}_{{\Omega }_1}=f(z)$ for $f=Ug$ becomes

$$(Ug)(z)=J_{\Phi }(z)g(\Phi (z))=⟨Ug,k_z^{(1)}{⟩}_{{\Omega }_1}=⟨g,U^{\ast }{k}_z^{(1)}{⟩}_{{\Omega }_2}.$$

On the other hand $g(\Phi (z))=⟨g,k_{\Phi (z)}^{(2)}{⟩}_{{\Omega }_2}$, so $J_{\Phi }(z)⟨g,k_{\Phi (z)}^{(2)}⟩=⟨g,U^{\ast }{k}_z^{(1)}⟩$ for all $g$, giving $U^{\ast }{k}_z^{(1)}=\overline{J_{\Phi }(z)}{k}_{\Phi (z)}^{(2)}$. Since $U$ is unitary, $U{U}^{\ast }=\mathrm{id}$, and applying $U$:

$$k_z^{(1)}=U\left(\overline{J_{\Phi }(z)}{k}_{\Phi (z)}^{(2)}\right),\text{i.e.}{K}_{{\Omega }_1}(\zeta ,z)=\overline{J_{\Phi }(z)}{J}_{\Phi }(\zeta )K_{{\Omega }_2}(\Phi (\zeta ),\Phi (z)).$$

Renaming $\zeta \mapsto z$, $z\mapsto w$ yields the stated transformation law.

Now restrict to the diagonal. With $|J_{\Phi }(z){|}^2=J_{\Phi }(z)\overline{J_{\Phi }(z)}$,

$$\log K_{{\Omega }_1}(z,z)=\log K_{{\Omega }_2}(\Phi (z),\Phi (z))+\log J_{\Phi }(z)+\log \overline{J_{\Phi }(z)},$$

valid locally where a branch of $\log J_{\Phi }$ exists. The last two terms are holomorphic and antiholomorphic respectively, so the mixed operator ${\partial }_{z_j}{\partial }_{{\bar{z}}_k}$ annihilates them. Applying ${\partial }_{z_j}{\partial }_{{\bar{z}}_k}$ and using the chain rule on the first term, with $w=\Phi (z)$ and $\partial w_p/\partial z_j=\partial {\Phi }_p/\partial z_j$,

$$g_{j\bar{k}}^{{\Omega }_1}(z)=\sum _{p,q}\frac{\partial {\Phi }_p}{\partial z_j}(z)g_{p\bar{q}}^{{\Omega }_2}(\Phi (z))\overline{\frac{\partial {\Phi }_q}{\partial z_k}(z)},$$

which is exactly the statement that $d{s}_{{\Omega }_1}^2={\Phi }^{\ast }(d{s}_{{\Omega }_2}^2)$. $\square$

The argument is Bergman's own (1933), reorganised by Range Ch. I §3 and Krantz §1.4 around the unitary-intertwiner viewpoint that makes the kernel transformation a one-line consequence of $U^{\ast }{k}^{(1)}=\overline{J}{k}^{(2)}$.

Bridge. The transformation law builds toward the boundary analysis that appears again in 06.10.09 (Szegő kernel and Fefferman asymptotics): once one knows the kernel transforms by Jacobian factors, the leading boundary singularity of $K_{\Omega }(z,z)$ on a strongly pseudoconvex domain becomes a biholomorphic invariant, and this is exactly what powers Fefferman's mapping theorem. The foundational reason the logarithm enters the metric is the cancellation just computed — the holomorphic Jacobian terms are killed by the mixed second derivative, so $\partial \bar{\partial }\log K$ is the invariant content. This pattern, that an $L^2$ minimization problem produces a reproducing kernel whose curvature is geometrically meaningful, is dual to the way the $\bar{\partial }$ $L^2$-theory of 06.10.04 produces minimal solutions; putting these together, the Bergman projection $L^2\to A^2$ and the $\bar{\partial }$-Neumann operator are two faces of the same orthogonal-decomposition machinery, and the bridge is the Hilbert-space geometry of holomorphic $L^2$ spaces. The central insight carried forward is that the kernel is a complete biholomorphic invariant of bounded domains up to the metric it induces.

Exercises Intermediate+

Lean formalization Intermediate+

Mathlib does not currently formalise the Bergman space $A^2(\Omega )$, the boundedness of point evaluation on it, or the reproducing kernel obtained from the Riesz representation theorem. A proposed signature, in Lean 4 / Mathlib syntax, sketching the target statement:

-- Sketch only; no current Mathlib coverage. See lean_mathlib_gap.
import Mathlib.Analysis.InnerProductSpace.Basic
import Mathlib.Analysis.Analytic.Basic

namespace Codex.RiemannSurfaces.SeveralComplexVariables

variable {n : ℕ} (Ω : Set (Fin n → ℂ)) (hΩ : IsOpen Ω) (hb : Bornology.IsBounded Ω)

-- The Bergman space: square-integrable holomorphic functions on Ω.
structure BergmanSpace where
  toFun : (Fin n → ℂ) → ℂ
  holo : HolomorphicOn toFun Ω
  l2   : MemLp toFun 2 (volume.restrict Ω)

-- Point evaluation is a bounded linear functional (interior estimate),
-- so Riesz representation yields the reproducing kernel.
theorem bergman_reproducing (z : Fin n → ℂ) (hz : z ∈ Ω) :
    ∃ k : BergmanSpace Ω, ∀ f : BergmanSpace Ω,
      f.toFun z = innerProductL2 f.toFun k.toFun := by
  sorry

end Codex.RiemannSurfaces.SeveralComplexVariables

The proof depends on names absent from Mathlib: the closedness of $A^2(\Omega )$ inside $L^2$ via the interior Cauchy estimate, the boundedness of ${\mathrm{ev}}_z$, and the integral-kernel form of the orthogonal projection. Each is a candidate Mathlib contribution; until then this unit ships with lean_status: none.

Advanced results Master

Series representation and extremal characterisation. For any orthonormal basis $\{{\varphi }_j\}$ of $A^2(\Omega )$, the kernel is $K_{\Omega }(z,w)={\sum }_j{\varphi }_j(z)\overline{{\varphi }_j(w)}$, converging locally uniformly on $\Omega \times \Omega$. The diagonal value carries an extremal reading: $K_{\Omega }(z,z)=\sup \{|f(z){|}^2:f\in A^2(\Omega ),\Vert f\Vert \le 1\}$, the supremum attained by $f=K_{\Omega }(\cdot ,z)/K_{\Omega }(z,z{)}^{1/2}$. This extremal description is independent of any basis and is the form Bergman used in 1933 to define the kernel function directly. It exhibits $K_{\Omega }(z,z)$ as a monotone-decreasing function of the domain: ${\Omega }_1\subseteq {\Omega }_2$ forces $K_{{\Omega }_1}(z,z)\ge K_{{\Omega }_2}(z,z)$, because the larger domain imposes the larger normalising integral on competitors.

The Bergman projection. Orthogonal projection $P:L^2(\Omega )\to A^2(\Omega )$ is the integral operator $(Pf)(z)={\int }_{\Omega }{K}_{\Omega }(z,\zeta )f(\zeta )dV(\zeta )$. The mapping properties of $P$ on Sobolev and Hölder scales — Condition $R$, the requirement that $P$ preserve $C^{\infty }(\overline{\Omega })$ — are the analytic crux of boundary regularity for biholomorphisms. On strongly pseudoconvex and on smoothly bounded pseudoconvex domains of finite type, $P$ is known to satisfy Condition $R$; the worm domains of Diederich-Fornæss show that Condition $R$ can fail on smoothly bounded pseudoconvex domains, so the hypothesis is genuine. The Bergman projection is the orthogonal-decomposition partner of the $\bar{\partial }$-Neumann operator: $P=I-{\bar{\partial }}^{\ast }N\bar{\partial }$ on the appropriate space, tying the kernel directly to the $L^2$-theory of 06.10.04.

Fefferman boundary asymptotics. For $\Omega$ smoothly bounded and strongly pseudoconvex with defining function $\rho <0$ inside, Fefferman 1974 proved the asymptotic expansion

$$K_{\Omega }(z,z)=\frac{\phi (z)}{(-\rho (z){)}^{n+1}}+\psi (z)\log (-\rho (z)),$$

with $\phi ,\psi \in C^{\infty }(\overline{\Omega })$ and $\phi >0$ on the boundary. The leading term recovers, near each boundary point, the ball kernel's $(1-|z{|}^2{)}^{-(n+1)}$ singularity up to the local CR-geometry of $\partial \Omega$. The expansion is the engine of Fefferman's mapping theorem: a biholomorphism between smoothly bounded strongly pseudoconvex domains extends to a diffeomorphism of the closures, because the kernel transformation law forces the boundary singularities to match, and the matching propagates inward through the $\bar{\partial }$-Neumann estimates. The logarithmic term is a genuine obstruction to $C^{\infty }$ behaviour of $K$ up to the boundary and is the origin of the parabolic-invariant theory of Fefferman and the CR $Q$-curvature.

Bergman completeness and representative coordinates. A bounded pseudoconvex domain is Bergman complete: the Bergman metric is a complete Riemannian metric (Kobayashi; Ohsawa). At a point $z_0$, Bergman's representative coordinates — built from ${\partial }_w\log (K(z,w)/K(w,w))$ evaluated at $z_0$ — linearise the automorphism group fixing $z_0$ and exhibit the metric as Euclidean to second order, the SCV analogue of normal coordinates. On the ball these coordinates are global and the metric is the complex hyperbolic metric of constant holomorphic sectional curvature, recovering the maximal symmetry $\mathrm{Aut}({\mathbb{B}}^n)=\mathrm{PU}(n,1)$.

Lu Qi-Keng problem and zeros of the kernel. Whether $K_{\Omega }(z,w)$ is zero-free on $\Omega \times \Omega$ — the Lu Qi-Keng problem — is false in general: explicit bounded domains (annuli in $\mathbb{C}$, certain Hartogs domains) have kernels with zeros. Zero-freeness is needed for the global representative-coordinate construction, so its failure marks a real obstruction. The ball and polydisc are Lu Qi-Keng domains by their explicit product/negative-binomial formulas; small perturbations can destroy the property.

Synthesis. The Bergman kernel is the reproducing kernel of the Hilbert space $A^2(\Omega )$, and this single $L^2$-extremal object organises the whole metric and boundary theory of bounded domains. The transformation law identifies the kernel with a biholomorphic invariant: putting the law together with the diagonal extremal characterisation, the kernel's growth and zeros are intrinsic, and the metric $\partial \bar{\partial }\log K$ is exactly the invariant Kähler structure that a biholomorphism must preserve. This is the foundational reason the ball and polydisc are distinguished — their kernels carry incompatible curvature, and curvature is an isometry invariant. The boundary expansion is dual to the interior extremal problem: Fefferman's $(-\rho {)}^{-(n+1)}$ singularity is the boundary shadow of the same minimization that defines $K(z,z)$ in the interior, and it generalises the ball computation to every strongly pseudoconvex domain. The central insight is that one orthogonal projection $P:L^2\to A^2$ encodes simultaneously the value-recovery (reproducing identity), the geometry (Bergman metric), the rigidity (Fefferman mapping), and the link to the $\bar{\partial }$-Neumann operator of 06.10.04; the Bergman theory is the statement that these are facets of a single Hilbert-space construction, and the boundary asymptotics are what convert the interior $L^2$-data into the smooth-mapping rigidity of strongly pseudoconvex domains.

Full proof set Master

Proposition (closedness of $A^2(\Omega )$ and the interior estimate). Let $\Omega \subseteq {\mathbb{C}}^n$ be open. Then $A^2(\Omega )$ is a closed subspace of $L^2(\Omega )$, and for every compact $L\subset \Omega$ there is $C_L$ with ${\sup }_L|f|\le C_L\Vert f\Vert$ for all $f\in A^2(\Omega )$.

Proof. Fix $z\in \Omega$ and $r$ with $\overline{B(z,r)}\subset \Omega$. Holomorphy gives the sub-mean-value property for $|f{|}^2$ (it is plurisubharmonic 06.10.03): $|f(z){|}^2\le \frac{1}{V(B(z,r))}{\int }_{B(z,r)}|f{|}^{2}dV$, where $V(B(z,r))={\pi }^n{r}^{2n}/n!$. Hence $|f(z)|\le V(B(z,r){)}^{-1/2}\Vert f{\Vert }_{L^2(B(z,r))}\le V(B(z,r){)}^{-1/2}\Vert f\Vert$. For a compact $L$, choose $r$ less than the distance from $L$ to $\partial \Omega$; the constant $C_L=({\pi }^n{r}^{2n}/n!{)}^{-1/2}$ works uniformly on $L$. Now let $f_k\in A^2(\Omega )$ converge to $g$ in $L^2(\Omega )$. The estimate applied to $f_k-f_m$ shows $\{f_k\}$ is uniformly Cauchy on each compact set, so it converges locally uniformly to a holomorphic limit $h$ (Weierstrass, several variables 06.07.01). Locally uniform and $L^2$ limits agree almost everywhere, so $g=h\in A^2(\Omega )$. Thus $A^2(\Omega )$ is closed. $\square$

Proposition (existence, uniqueness, and symmetry of the kernel). For $\Omega$ with $A^2(\Omega )\ne \{0\}$, there is a unique $K_{\Omega }:\Omega \times \Omega \to \mathbb{C}$, holomorphic in the first and antiholomorphic in the second argument, with $f(z)={\int }_{\Omega }f(\zeta )\overline{K_{\Omega }(\zeta ,z)}dV(\zeta )$ for all $f\in A^2$ and all $z$; moreover $K_{\Omega }(z,w)=\overline{K_{\Omega }(w,z)}$.

Proof. By the previous proposition ${\mathrm{ev}}_z$ is bounded on the Hilbert space $A^2(\Omega )$, so Riesz representation supplies a unique $k_z\in A^2(\Omega )$ with $f(z)=⟨f,k_z⟩$ for all $f$. Set $K_{\Omega }(z,w):=k_w(z)=⟨k_w,k_z⟩$. Conjugate symmetry is the Hermitian symmetry of the inner product: $K_{\Omega }(z,w)=⟨k_w,k_z⟩=\overline{⟨k_z,k_w⟩}=\overline{K_{\Omega }(w,z)}$. Holomorphy in $z$ is holomorphy of $k_w$; antiholomorphy in $w$ follows from the symmetry. Uniqueness: if $\tilde{K}$ also reproduces, then for each $z$ the element $\tilde{K}(\cdot ,z)$ represents ${\mathrm{ev}}_z$, which Riesz makes unique, so $\tilde{K}(\cdot ,z)=K_{\Omega }(\cdot ,z)$. $\square$

Proposition (positivity and the extremal value). For $\Omega$ bounded, $K_{\Omega }(z,z)=\sup \{|f(z){|}^2:\Vert f\Vert \le 1\}>0$.

Proof. For $\Vert f\Vert \le 1$, Cauchy-Schwarz gives $|f(z)|=|⟨f,k_z⟩|\le \Vert k_z\Vert =K_{\Omega }(z,z{)}^{1/2}$, with equality at $f=k_z/\Vert k_z\Vert$, so the supremum equals $K_{\Omega }(z,z)$. Positivity: constants lie in $A^2(\Omega )$ since $\Omega$ is bounded, and the constant $1/\Vert 1\Vert$ has $|f(z){|}^2=1/V(\Omega )>0$, forcing the supremum to be positive. $\square$

Proposition (the metric is Hermitian positive definite for bounded $\Omega$). For $\Omega$ bounded, the matrix $g_{j\bar{k}}(z)={\partial }_{z_j}{\partial }_{{\bar{z}}_k}\log K_{\Omega }(z,z)$ is Hermitian and positive definite at each $z\in \Omega$.

Proof. Hermitian symmetry is immediate from conjugating and swapping $j,k$. For positive definiteness fix $z$ and a vector $\xi \in {\mathbb{C}}^n$; the quantity ${\sum }_{j,k}{g}_{j\bar{k}}(z){\xi }_j{\bar{\xi }}_k$ is computed from $\log K(z,z)$. Writing $K={\sum }_j|{\varphi }_j{|}^2$ on the diagonal and using the Cauchy-Schwarz / Lagrange-identity expansion of $\partial \bar{\partial }\log K=(K\partial \bar{\partial }K-\partial K\wedge \bar{\partial }K)/K^2$, the numerator is the Gram determinant of the vectors $({\varphi }_j(z),\nabla {\varphi }_j(z))$ paired against $\xi$, which is non-negative and strictly positive unless all directional derivatives ${\sum }_j\overline{{\varphi }_j(z)}\xi \cdot \nabla {\varphi }_j(z)$ are proportional to the values ${\varphi }_j(z)$. Because $A^2(\Omega )$ contains, for bounded $\Omega$, functions separating points and giving independent first-order jets (polynomials restrict into $A^2$), no non-zero $\xi$ makes the form vanish, so $(g_{j\bar{k}})$ is positive definite. $\square$

Theorem (transformation law). Statement and proof as in the Intermediate-tier Key theorem section.

Proof. The unitary intertwiner $U:A^2({\Omega }_2)\to A^2({\Omega }_1)$, $(Uf)(z)=J_{\Phi }(z)f(\Phi (z))$, is an isometry by the holomorphic change of variables $dV\circ \Phi =|J_{\Phi }{|}^{2}dV$; carrying reproducing kernels across $U$ gives $K_{{\Omega }_1}(z,w)=J_{\Phi }(z)K_{{\Omega }_2}(\Phi (z),\Phi (w))\overline{J_{\Phi }(w)}$, and the mixed second derivative of the logarithm kills the holomorphic Jacobian factors, yielding the tensorial invariance of $d{s}^2$. The detailed computation is given in the Intermediate section. $\square$

Connections Master

• Holomorphic functions of several variables 06.07.01. The Bergman space $A^2(\Omega )$ is carved out of the holomorphic functions of $n$ variables; the interior Cauchy estimate that makes point evaluation bounded — and hence makes the kernel exist — is the several-variable Weierstrass convergence theorem applied to $L^2$-Cauchy sequences. Every property of the kernel is downstream of the basic function theory established there.

• Pseudoconvexity and the Levi form 06.10.03. Boundedness alone gives a kernel, but the deepest theorems — Bergman completeness, the smoothness of the Bergman projection, Fefferman's boundary expansion — hold on (strongly) pseudoconvex domains. The plurisubharmonicity of $|f{|}^2$ used in the interior estimate, and the Levi geometry of the boundary controlling the kernel singularity, both belong to this unit's circle of ideas.

• The $\bar{\partial }$-equation and Hörmander's $L^2$ estimates 06.10.04. The Bergman projection $P:L^2\to A^2$ and the $\bar{\partial }$-Neumann operator are partners in the orthogonal decomposition $L^2=A^2\oplus \overline{\mathrm{ran}{\bar{\partial }}^{\ast }}$; the identity $P=I-{\bar{\partial }}^{\ast }N\bar{\partial }$ ties the kernel directly to the minimal-solution theory, and the regularity of $P$ is read off the $\bar{\partial }$-Neumann estimates.

• Domains of holomorphy and holomorphic convexity 06.10.01. The monotonicity $K_{{\Omega }_1}\ge K_{{\Omega }_2}$ for ${\Omega }_1\subseteq {\Omega }_2$ and the boundary blow-up of $K$ make the diagonal kernel an exhaustion-type invariant; on a domain of holomorphy the kernel and metric are complete, reflecting that the domain cannot be holomorphically enlarged.

• Szegő kernel and Fefferman asymptotics 06.10.09. The Szegő kernel is the boundary $H^2$ analogue of the interior $A^2$ Bergman kernel; Fefferman's asymptotic expansion of $K_{\Omega }(z,z)$ near a strongly pseudoconvex boundary, and the resulting smooth extension of biholomorphisms, are developed there using the transformation law proved in this unit.

Historical & philosophical context Master

Stefan Bergman introduced the kernel function in two 1922 dissertations and the foundational papers Über die Kernfunktion eines Bereiches und ihr Verhalten am Rande of 1933 ^{[Bergman 1933]} (J. Reine Angew. Math. 169 and 172), where he defined $K_{\Omega }$ by the extremal property $K_{\Omega }(z,z)=\sup \{|f(z){|}^2:\Vert f\Vert \le 1\}$ and studied its boundary behaviour. His 1950 monograph The Kernel Function and Conformal Mapping ^{[Bergman 1950]} (AMS Mathematical Surveys 5) consolidated the reproducing-kernel formalism and the invariant metric now bearing his name; the metric had appeared in his work and in parallel investigations by Salomon Bochner. The reproducing-kernel apparatus itself was placed on its abstract footing by Nachman Aronszajn's 1950 Theory of reproducing kernels (Trans. Amer. Math. Soc. 68), which subsumes the Bergman and Szegő kernels as instances of a Hilbert space of functions with bounded point evaluations.

The subject's modern depth came from Charles Fefferman's 1974 The Bergman kernel and biholomorphic mappings of pseudoconvex domains ^{[Fefferman 1974]} (Invent. Math. 26, 1-65), which established the boundary asymptotic expansion of $K_{\Omega }(z,z)$ on smoothly bounded strongly pseudoconvex domains and used it to prove that biholomorphisms extend smoothly to the boundary — answering a question open since Henri Poincaré's 1907 observation that the ball and bidisc in ${\mathbb{C}}^2$ are not biholomorphic. Fefferman's parametrix for the Bergman kernel, refined by Boutet de Monvel-Sjöstrand for the Szegő kernel, opened the parabolic-invariant theory of CR boundaries and the study of CR $Q$-curvature. The transformation law and the explicit ball and polydisc kernels treated here are the elementary core on which that boundary theory is built; Krantz's Function Theory of Several Complex Variables Ch. 7 ^{[Krantz Ch. 7]} and his later monograph Geometric Analysis of the Bergman Kernel and Metric ^{[Krantz GTM 268]} present the line from Bergman's extremal definition to Fefferman's asymptotics as a single development.

Bibliography Master

@article{Bergman1933Kernfunktion,
  author  = {Bergman, Stefan},
  title   = {{\"U}ber die Kernfunktion eines Bereiches und ihr Verhalten am Rande. I},
  journal = {J. Reine Angew. Math.},
  volume  = {169},
  year    = {1933},
  pages   = {1--42}
}

@book{Bergman1950Kernel,
  author    = {Bergman, Stefan},
  title     = {The Kernel Function and Conformal Mapping},
  series    = {Mathematical Surveys},
  number    = {5},
  publisher = {American Mathematical Society},
  year      = {1950}
}

@article{Aronszajn1950RKHS,
  author  = {Aronszajn, Nachman},
  title   = {Theory of reproducing kernels},
  journal = {Trans. Amer. Math. Soc.},
  volume  = {68},
  year    = {1950},
  pages   = {337--404}
}

@article{Fefferman1974Bergman,
  author  = {Fefferman, Charles},
  title   = {The {B}ergman kernel and biholomorphic mappings of pseudoconvex domains},
  journal = {Invent. Math.},
  volume  = {26},
  year    = {1974},
  pages   = {1--65}
}

@book{KrantzFTSCV,
  author    = {Krantz, Steven G.},
  title     = {Function Theory of Several Complex Variables},
  edition   = {2},
  series    = {AMS Chelsea Publishing},
  volume    = {340},
  publisher = {American Mathematical Society},
  year      = {2001}
}

@book{KrantzBergman,
  author    = {Krantz, Steven G.},
  title     = {Geometric Analysis of the Bergman Kernel and Metric},
  series    = {Graduate Texts in Mathematics},
  volume    = {268},
  publisher = {Springer},
  year      = {2013}
}

@book{RangeHolomorphic,
  author    = {Range, R. Michael},
  title     = {Holomorphic Functions and Integral Representations in Several Complex Variables},
  series    = {Graduate Texts in Mathematics},
  volume    = {108},
  publisher = {Springer},
  year      = {1986}
}

@article{DiederichFornaess1977Worm,
  author  = {Diederich, Klas and Forn{\ae}ss, John Erik},
  title   = {Pseudoconvex domains: an example with nontrivial Nebenh{\"u}lle},
  journal = {Math. Ann.},
  volume  = {225},
  year    = {1977},
  pages   = {275--292}
}