Real-Analytic Kähler Potentials

• Real-analytic Kähler potentials are defined via convergent power series in holomorphic and conjugate variables that uniquely determine the Kähler metric through i∂∂̄ϕ.
• Their analytic nature enables precise Bergman kernel expansions with factorial bounds and supports exponential accuracy in quantization schemes.
• The polarization of these potentials yields explicit geometric invariants and rigidity results, facilitating advanced studies in complex and algebraic geometry.

A real-analytic Kähler potential is a real-valued function defined in a holomorphic coordinate chart of a complex manifold whose power series expansion in the complex and conjugate variables converges in a neighborhood of the diagonal. Real-analytic Kähler potentials play a central role in Kähler geometry, admitting unique holomorphic extensions (polarizations) in mixed variables and underpinning a wide range of analytic, geometric, and quantization-theoretic results. The analyticity of these potentials facilitates properties that are not attainable for merely smooth ($C^\infty$) potentials, including sharper Bergman kernel expansions, reflection principles, rigidity phenomena, and quantization schemes.

1. Definition and Local Characterization

Let $M$ be a complex $n$-dimensional manifold and $\omega$ a Kähler form on $M$. By definition, a real-analytic Kähler potential $\varphi$ is a real-valued, real-analytic function in a holomorphic coordinate chart $U\subset M$ such that

$\omega = i\,\partial\bar\partial\varphi,$

with $\varphi: U \to \mathbb{R}$. The analyticity implies the existence of a convergent power series

$$\varphi(z,\bar z) = \sum_{|\alpha|,|\beta|\ge 0} a_{\alpha,\beta}\,z^\alpha\bar z^\beta$$

in a bidisk neighborhood of each point. This facilitates the extension of $M$0 to a holomorphic function $M$1 (the polarization) satisfying $M$2 and $M$3 in the domain of convergence (Hezari et al., 2017, Cheng et al., 2020).

2. Polarization and Diastasis

For a real-analytic Kähler potential $M$4, the polarization $M$5 is holomorphic in $M$6 and antiholomorphic in $M$7, and it encodes the two-point geometry of the metric. The diastatic function, first introduced by Calabi, is defined as

$M$8

which is real-valued, vanishes at $M$9, and determines local isometric invariants. In important cases, such as complex space forms and Hartogs domains, the polarization and the diastasis admit explicit rational or Nash-algebraic forms (Cheng et al., 2020). The analyticity of the potential ensures strong regularity and algebraic properties for these invariants.

3. Analyticity and Functional Equivalence

A pivotal result is that analyticity of the Kähler potential is characterized by functional-analytic properties in infinite-dimensional geometry. In the context of the Mabuchi space of Kähler potentials

$n$0

real-analyticity is both necessary and sufficient for the existence of symmetric involutive isometries (“complex Legendre symmetries”). Given any such symmetry $n$1 with $n$2 and $n$3, the fixed point $n$4 must be real-analytic (Lempert, 2017). The analytic wave-front set vanishes identically, and the reflection principle ensures holomorphic extendability across analytic arcs in the boundary-data setup of Donaldson's WZW equation.

4. Bergman Kernel Expansions and Quantization

Real-analytic Kähler potentials are distinguished by their effect on the asymptotics of Bergman kernels associated to powers of holomorphic line bundles. In the real-analytic case, the Bergman kernel $n$5 has an off-diagonal expansion of the form

$n$6

where $n$7 is the holomorphic polarization of $n$8, and the coefficients $n$9 obey strong factorial bounds $\omega$0. The remainder $\omega$1 is of factorial type, and in the fully analytic setting, the expansion persists in a neighborhood of size $\omega$2 (Hezari et al., 2017). For metrics with locally constant holomorphic sectional curvature, the expansion holds in a fixed neighborhood with exponentially small remainder, and the amplitudes can be given explicitly (Hezari et al., 2017, Hezari et al., 2019).

In contrast, for merely $\omega$3 potentials, the off-diagonal expansion is valid only in a smaller neighborhood ($\omega$4), and there is no factorial control of the coefficients (Hezari et al., 2017, Hezari et al., 2018).

In quantization theory, the analyticity ensures that the Berezin–Toeplitz calculus and quantized geodesics in the Mabuchi space exhibit exponentially accurate symbolic behavior, underpinned by analytic Fourier-Integral Operator (FIO) methods (Deleporte et al., 2022). The convergence of “quantized geodesics” (Bergman geodesics) to classical geodesics is established at $\omega$5 in the presence of analytic data.

5. Algebraicity, Nash Polarizations, and Rigidity

When the polarization $\omega$6 of a real-analytic Kähler potential is Nash-algebraic, i.e., satisfies a nontrivial polynomial equation in $\omega$7, there are strong rigidity results. Specifically, such manifolds cannot be relatives (in the sense of sharing a nontrivial holomorphic isometric submanifold) of any complex space form of constant curvature (flat, projective, or hyperbolic spaces) (Cheng et al., 2020). This extends Calabi’s rigidity results and ensures the algebraic independence of Kähler geometries under certain analytic and algebraic constraints.

Diastatic-function polarizations also imply obstructions to relativity, especially in the context of Hartogs domains and spaces constructed as total spaces over Kähler bases. The fiber–base splitting in the diastasis prevents the possibility of common Kähler submanifolds with space forms if the base is already non-relative.

6. Construction and Practical Schemes for Analytic Potentials

Explicit analytic expressions for Kähler potentials on complex varieties, including Calabi–Yau manifolds, can be constructed by combining algebraic machinery with numerical and machine learning techniques. Analytic Ansätze for the potential $\omega$8 are formulated in terms of Fubini–Study terms plus finite-dimensional expansions in homogeneous coordinates, with moduli-dependent coefficients determined via symbolic regression on neural network outputs (Constantin et al., 12 Mar 2026). Accuracy at the percent level has been achieved in examples with $\omega$9, and explicit closed-form expressions have been extracted for the moduli-dependent coefficients, respecting the underlying discrete symmetries.

The combination of real-analytic structure, polarizations, and explicit algebraic description enables new approaches to the construction, approximation, and understanding of Ricci-flat Kähler metrics on Calabi–Yau spaces, facilitating advances in both geometric analysis and string-theoretic applications (Constantin et al., 12 Mar 2026).

7. Analyticity in Geodesic Flows and Boundary Problems

The initial value problem for geodesics in the Mabuchi space of real-analytic Kähler potentials admits a unique real-analytic solution for short time intervals, established via the Cauchy–Kovalevskaya theorem. Analyticity guarantees short-time existence and the convergence of quantized and classical geodesic flows (Deleporte et al., 2022). However, for the boundary value problem, analytic solutions almost never exist for generic endpoint data: analytic geodesics correspond to autonomous Hamiltonian flows, which are non-generic among all time-dependent flows. Thus, generically, the existence of boundary-value analytic geodesics is precluded in the analytic category (Deleporte et al., 2022).

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In summary, real-analytic Kähler potentials constitute a distinguished regularity class with far-reaching implications in Kähler geometry, global analysis, quantization, and rigidity theory. Their properties underpin enhanced analytic control, strong asymptotic estimates, and comprehensive geometric rigidity, distinguishing them sharply from the merely smooth category. Key advances rely on the interplay of polarization, analytic microlocal analysis, functional and geometric quantization, and algebraic geometry, as elucidated in recent work (Lempert, 2017, Hezari et al., 2017, Hezari et al., 2019, Hezari et al., 2018, Constantin et al., 12 Mar 2026, Deleporte et al., 2022, Cheng et al., 2020).