Holomorphic Inner Product in Complex Geometry

• Holomorphic inner products are Hermitian sesquilinear pairings defined on spaces of holomorphic functions and sections, connecting complex geometry and functional analysis.
• Canonical constructions, such as L²-pairings on Kähler manifolds and weighted Bergman inner products on bounded symmetric domains, offer concrete methodologies and quantization insights.
• Applications span representation theory, automorphic forms, and geometric invariant theory, with explicit formulations aiding in norm computations and branching laws.

A holomorphic inner product is a Hermitian sesquilinear pairing on a space of holomorphic functions (or sections), fundamentally applied across complex geometry, representation theory, and the theory of automorphic forms. In modern mathematical contexts, the term encompasses both concrete constructions (e.g., $L^2$-pairings of sections of a line bundle over a Kähler manifold, Petersson inner products on modular forms, and weighted Bergman pairings on bounded symmetric domains) and structural results describing the interplay between infinite-dimensional and finite-dimensional objects via such pairings. This article details the definitions, canonical constructions, surjectivity/injectivity phenomena, and application spectra of holomorphic inner products with rigorous mathematical formulations and references to precise sources.

1. Canonical Constructions of Holomorphic Inner Products

The archetype of a holomorphic inner product arises in the setting of a compact Kähler manifold $X$ of complex dimension $n$, equipped with a very ample holomorphic line bundle $\mathcal{L}$, and its space of global holomorphic sections $H^0(X, \mathcal{L})$. Given a smooth Hermitian metric $h$ on $\mathcal{L}$ with positive curvature form $$\omega_h = -\frac{\sqrt{-1}}{2\pi} \partial \bar{\partial} \log h$$ representing $c_1(\mathcal{L})$, the $L^2$-inner product on $H^0(X, \mathcal{L})$ is

$$\langle s, t \rangle_{Hilb(h)} = \int_X h(s(x), t(x))\, \frac{\omega_h^n}{n!},$$

where $s, t \in H^0(X, \mathcal{L})$. This construction is functorial in the sense that it defines a map, the Hilbert map,

$$\mathrm{Hilb} : \mathcal{H}(X, \mathcal{L}) \to \mathcal{B},$$

where $\mathcal{H}(X, \mathcal{L})$ is the space of smooth Hermitian metrics with positive curvature and $\mathcal{B}$ is the finite-dimensional symmetric space of positive-definite Hermitian forms on $H^0(X, \mathcal{L})$ (identifiable with the space of $N \times N$ Hermitian matrices for $N = \dim H^0(X, \mathcal{L})$) (Hashimoto, 2017).

On bounded symmetric domains $D \subset \mathfrak{p}^+$ associated to a Hermitian symmetric space $G/K$, the weighted Bergman inner product for the holomorphic discrete series is

$$\langle f, g \rangle_\lambda = C_\lambda \int_D f(x)\, \overline{g(x)}\, h(x, \bar{x})^{\lambda - p}\, dx,$$

where $h(x, \bar{x})$ is the generic norm, $p = 2n/r$ is the genus (with $n = \dim_\mathbb{C}\mathfrak{p}^+$, $r = \mathrm{rank}$), and $C_\lambda$ is a normalization constant. This structure realizes a unitary highest-weight representation for $\lambda > p-1$ (Nakahama, 2021).

2. Representation, Surjectivity, and Classification Results

A key structural theorem for holomorphic inner products is the surjectivity of the Hilbert map for very ample line bundles: for any positive-definite Hermitian form $G$ on $H^0(X, \mathcal{L})$, there exists a Hermitian metric $h$ on $\mathcal{L}$ such that

$$G(s, t) = \int_X h(s(x), t(x)) \frac{\omega_h^n}{n!}, \quad \forall s, t \in H^0(X, \mathcal{L}).$$

Equivalently, every positive-definite Hermitian form on $H^0$ can be realized as an $L^2$-inner product induced from some metric $h$ (Hashimoto, 2017). This is accomplished via a diffeomorphic "Hilbert–Fubini–Study" correspondence, leveraging the geometry of projective space and degree theory on positive Hermitian matrices, as well as via a PDE reduction to scalar equations involving potentials $e^{\phi}$ and the Monge–Ampère equation.

The converse construction (the Fubini–Study map) associates to each $H\in\mathcal{B}$ a Hermitian metric $h_{FS(H)}$ on $\mathcal{L}$ via

$$\sum_{i=1}^N |s_i(x)|_{h_{FS(H)}}^2 = 1\quad\forall x\in X,$$

for any $H$-orthonormal basis $\{s_i\}$, and this map is injective: $FS(H)=FS(H')$ implies $H=H'$ (Hashimoto, 2017).

For holomorphic discrete series on bounded symmetric domains, the inner product structure enables explicit decomposition of representation spaces under restriction to subgroups via the branching law of Hua–Kostant–Schmid–Kobayashi, and norm computations reduce to explicit hypergeometric expansions (Nakahama, 2021).

3. Explicit Formulas: Modular and Automorphic Forms

In the theory of modular and automorphic forms, holomorphic inner products admit concrete realization. The Petersson inner product for Siegel modular forms (for instance, in the vector-valued case of Siegel genus 2) is

$$\langle F_1, F_2 \rangle = \int_{\Gamma_0^{(2)}(N) \backslash \mathfrak{h}_2} \langle F_1(Z), F_2(Z) \rangle_\kappa\, (\det Y)^{\kappa_1+2}\, d\mu(Z).$$

In the context of theta lifts (Yoshida lifts), the explicit inner product formula connects the Petersson norm to $L$-values: $$\langle\theta_f^*,\theta_f^*\rangle \cdot \|\phi_f^D\|_R^{-1} = N 2^\beta (2k_1+1)^{-1} (2k_2+1)^{-1} L(1, As^+(\pi)) \prod_{p|N}(1+\epsilon_p) \prod_{p|\Delta_F} (1+p^{-1}),$$ where all factors (dimension, embedding norm, special $L$-values, local Euler factors) are explicitly computable (Hsieh et al., 2016).

For holomorphic discrete series on bounded symmetric domains $D$, formulas for inner products of polynomials against exponential reproducing kernels involve the Gindikin gamma-factor, multivariate Pochhammer symbols, and multivariate hypergeometric functions of type $BC$ (Nakahama, 2021). In rank-one (classical) cases, these reduce to classical Gauss hypergeometric integrals.

4. Orthogonality, Degeneracy, and Extensions

On spaces of holomorphic forms with logarithmic growth or singularities, the naive $L^2$-pairing typically diverges. The regularized inner product construction for weakly holomorphic modular forms of weight $2k$ employs analytic continuation and damping factors: $$(f,g) := \underset{s=0}{\mathrm{CT}}\, I_0(f,g;0,s) - \sum_{n>0} c_f(-n) c_g(-n) \Im (E_{2-2k}(-4\pi n)),$$ where $I_0(f,g;w,s)$ is the regularized integral, and $E_\nu(x)$ is a generalized exponential integral (1711.01733). The radical (nullspace) of this pairing is exactly the image of repeated differentiation $D^{2k-1}$ applied to weak cusp forms: $M_{2k}^\perp = D^{2k-1}\left(S_{2-2k}\right),$ where $S_{2-2k}$ denotes the space of weak cusp forms of weight $2-2k$. This correspondence underpins the congruence relations and the algebraic structure of Hecke eigenforms in the regularized inner product setting.

5. Moment Map, Quantization, and Geometric Invariant Theory Perspectives

The Hilbert and Fubini–Study maps exemplify finite-dimensional reductions of infinite-dimensional geometric data, serving as a moment map for the $U(N)$-action on projective space. Surjectivity of the Hilbert map equates to properness of the moment map, ensuring the existence of zeros, while injectivity of the Fubini–Study map corresponds to uniqueness of critical points (Hashimoto, 2017).

In the large $k$ limit (high powers of the line bundle), the compositions

$FS_k \circ Hilb_k, \quad Hilb_k \circ FS_k$

tend to the identity as $k\to\infty$, mirroring the semi-classical regime in Berezin–Toeplitz quantization of Kähler metrics, and undergirding asymptotic analytic and geometric quantization.

6. Applications and Implications Across Fields

The classification of holomorphic inner products informs several advanced research areas:

• Balanced embeddings: The condition $FS(Hilb(h))=h$ for a metric $h$ corresponds to balanced metrics, unique up to automorphism (Hashimoto, 2017).
• Theta correspondence: Explicit inner product formulas relate special values of $L$-functions to norm computations for automorphic forms, supporting arithmetic non-vanishing theorems and congruence phenomena (Hsieh et al., 2016).
• Representation theory: Explicit weighted Bergman inner products permit discrete branching laws and symmetry breaking operator construction for holomorphic discrete series (Nakahama, 2021).
• Orthogonality and structure of modular forms: The radical structure of regularized holomorphic inner products clarifies the algebraic and analytic decomposition of modular form spaces, with applications to moonshine phenomena and string theory (1711.01733).
• Berezin–Toeplitz quantization: Surjectivity and injectivity results for Hilbert and Fubini–Study pairs are foundational in the quantization of Kähler manifolds (Hashimoto, 2017).

The body of work reviewed delineates a unified perspective on holomorphic inner products, connecting complex geometry, automorphic form theory, and representation theory by means of explicit analytic, algebraic, and geometric tools.