Diastasis Function in Kähler Geometry

• Diastasis function is a real-analytic, symmetric measure derived from the Kähler potential that locally approximates squared geodesic distances.
• It underpins criteria for holomorphic isometric immersions into complex space forms by enforcing Hermitian positivity in its power series expansion.
• The function governs Bergman kernel asymptotics and finds multidisciplinary applications in quantum field theory, information theory, and clinical anatomical modeling.

Calabi’s diastasis function is a canonical object in Kähler geometry that encodes the deviation of a Kähler potential from being a complex-bilinear quadratic form and has deep applications across complex geometry, geometric analysis, mathematical physics, and even clinical anatomy. It is characterized as a real-analytic, symmetric, and gauge-invariant function constructed from the analytic continuation of a Kähler potential, and, in both global and local contexts, provides a two-point analog of the local Kähler metric. The diastasis function not only underlies the structure of Kähler immersions into complex space forms but also serves as a central analytic potential governing asymptotics of Bergman kernels, interface conformal field theory entropies, and complexity measures of generalized coherent states.

1. Definition and Fundamental Properties

Let $(M, g)$ be a real-analytic Kähler manifold with local holomorphic coordinates $z = (z^1, ..., z^n)$ and a Kähler potential $\varphi(z, \bar z)$ such that $$\omega = \frac{i}{2} \partial \bar\partial \varphi(z, \bar z)$$ is the Kähler form. The real-analyticity of $\varphi$ guarantees that it can be analytically continued to a function $\varphi(z, \bar w)$ in a neighborhood of the diagonal of $M \times M$. Calabi's diastasis function $D(z, w)$ is defined by: $$D(z,w) = \varphi(z, \bar z) + \varphi(w, \bar w) - \varphi(z, \bar w) - \varphi(w, \bar z)$$ This function is symmetric ($D(z, w) = D(w, z)$), real-valued, vanishes on the diagonal ($z = (z^1, ..., z^n)$0), and is independent of the choice of Kähler potential due to the cancellation of holomorphic and anti-holomorphic ambiguities. The Taylor expansion in normal (Bochner) coordinates about a point $z = (z^1, ..., z^n)$1 yields $z = (z^1, ..., z^n)$2, meaning to second order, $z = (z^1, ..., z^n)$3 approximates the squared geodesic distance $z = (z^1, ..., z^n)$4 between $z = (z^1, ..., z^n)$5 and $z = (z^1, ..., z^n)$6 (Loi et al., 2017, Mossa, 2012). For small separation,

$z = (z^1, ..., z^n)$7

This property underpins the geometric interpretation of the diastasis function as a two-point generalization of the metric.

2. Analytic and Geometric Significance

The diastasis function is intrinsic to the metric and is the unique Kähler potential centered at a point $z = (z^1, ..., z^n)$8 whose power series expansion lacks purely holomorphic or anti-holomorphic terms. It always exists locally for real-analytic Kähler manifolds and may extend globally in special cases (e.g., Hermitian symmetric spaces, bounded symmetric domains, and certain noncompact settings) (Loi et al., 2017, Loi et al., 2016, Loi et al., 2009).

For Hermitian symmetric spaces, the diastasis function often admits an explicit global formula. For example, the diastasis for the complex hyperbolic ball $z = (z^1, ..., z^n)$9 with the Bergman (hyperbolic) metric is $\varphi(z, \bar z)$0 (Mossa, 2012). In these settings, the diastasis function is intimately linked to the spectral theory of the Laplacian and volume entropy, providing sharp lower bounds for the first eigenvalue and entropy in terms of the supremum of its gradient (Mossa, 2012).

On the Siegel–Jacobi disk, the function incorporates both the standard hyperbolic part and an additional non-symmetric component tied to the Heisenberg group structure, and serves as the logarithm of the normalized Berezin kernel (Berceanu, 2013).

3. Diastasis and Kähler Immersions

Calabi's diastasis function underlies necessary and sufficient conditions for the existence of (local) holomorphic isometric immersions of Kähler manifolds into complex space forms—that is, Kähler–Euclidean, projective (Fubini–Study), or hyperbolic spaces. The criterion operates by examining the sesquilinear expansion of the exponential of the diastasis (for flat case), or more generally, transforms involving $\varphi(z, \bar z)$1 for (non-flat) space forms of curvature $\varphi(z, \bar z)$2, and requiring positivity and finite rank of the corresponding Hermitian matrix (Loi et al., 2017).

Explicitly, for immersion into $\varphi(z, \bar z)$3, a neighborhood of a point $\varphi(z, \bar z)$4 immerses holomorphically if and only if the infinite Hermitian matrix $\varphi(z, \bar z)$5 from the power series $\varphi(z, \bar z)$6 is positive semidefinite of rank $\varphi(z, \bar z)$7. Analogous statements hold for projective and hyperbolic ambient spaces, replacing the diastasis power series by expansions in terms of $\varphi(z, \bar z)$8 (Loi et al., 2017). Non-immersion phenomena, such as the existence of metrics with globally positive diastasis but no Kähler immersion into any complex space form (e.g., the cigar soliton), illustrate the subtlety of high-order terms in the expansion (Loi et al., 2016).

4. Bergman/Calabi Diastasis and Bergman Kernel Asymptotics

The diastasis function arises naturally as the key factor controlling the off-diagonal decay of Bergman and Berezin kernels on compact Kähler or toric manifolds. For a positive line bundle with a real-analytic Hermitian metric $\varphi(z, \bar z)$9 over a compact Kähler manifold, the normalized Bergman kernel $$\omega = \frac{i}{2} \partial \bar\partial \varphi(z, \bar z)$$0 satisfies the asymptotic

$$\omega = \frac{i}{2} \partial \bar\partial \varphi(z, \bar z)$$1

as $$\omega = \frac{i}{2} \partial \bar\partial \varphi(z, \bar z)$$2 for $$\omega = \frac{i}{2} \partial \bar\partial \varphi(z, \bar z)$$3 near the diagonal, where $$\omega = \frac{i}{2} \partial \bar\partial \varphi(z, \bar z)$$4 is the diastasis (Zelditch, 2016). In the toric case, utilizing symplectic potential coordinates, the diastasis admits a global formula on real-positive slices (i.e., points on the same torus orbit), and exactly governs exponential decay rates of the kernel. For general $$\omega = \frac{i}{2} \partial \bar\partial \varphi(z, \bar z)$$5 metrics, this relation is only valid along certain orbits due to the absence of a globally defined analytic extension.

On bounded domains with Bergman metric of constant holomorphic sectional curvature, the so-called Bergman-Calabi diastasis can be expressed using the Bergman representative coordinate, and its behavior at the boundary of the domain characterizes hyperconvexity and biholomorphism to the Euclidean ball (Dong et al., 2021). This function also encodes the Lu Qi–Keng property and provides explicit control over the zero-set of the Bergman kernel.

5. Diastasis in Physical and Information-Theoretic Contexts

A remarkable aspect of Calabi's diastasis is its appearance in quantum field theory, quantum information, and mathematical physics, where it quantifies interface entropies, defect free energies, and complexity measures:

• Interface and defect entropy: In 2d $$\omega = \frac{i}{2} \partial \bar\partial \varphi(z, \bar z)$$6 SCFTs, the boundary $$\omega = \frac{i}{2} \partial \bar\partial \varphi(z, \bar z)$$7-function for BPS interfaces is precisely the exponential of the diastasis between two points in moduli space (Bachas et al., 2013, D'Hoker et al., 2014). In 4d $$\omega = \frac{i}{2} \partial \bar\partial \varphi(z, \bar z)$$8 SCFTs, the Janus interface entropy is shown to be proportional to the diastasis function on the conformal manifold of marginal couplings, $$\omega = \frac{i}{2} \partial \bar\partial \varphi(z, \bar z)$$9 with $\varphi$0 for supersymmetric settings (Goto et al., 2018, Goto et al., 2020). The defining four-term combination of Kähler potentials is uniquely suited for duality-invariant expressions in these contexts.
• Circuit complexity: In the context of generalized coherent states, such as for $\varphi$1 representations and the projective space $\varphi$2, the logarithmic circuit complexity for transforming one normalized coherent state to another is exactly the diastasis function—$\varphi$3—and only the Fubini–Study metric achieves this exact correspondence (Ray, 2024).
• Holography: In six-dimensional supergravity duals to 2d CFTs, the entanglement entropy across BPS interfaces and junctions is a sum of diastasis functions on the moduli space $\varphi$4, emphasizing the geometric nature of interface entropy as a function on a Kähler manifold (D'Hoker et al., 2014).

6. Applied Models: Diastasis Function in Clinical Anatomy

Beyond pure mathematics and quantum field theory, the term "diastasis function" has surfaced in computational anatomy, specifically in modeling the inter-rectus distance (IRD, or diastasis recti) along the linea alba in abdominal wall reconstructions. Based on 3D CT scan data, a patient-specific continuous function $\varphi$5 is constructed to model IRD as a function of normalized height $\varphi$6 and demographic covariates (Gueroult et al., 20 Jan 2025). This "diastasis function" is empirical, not Kähler-geometric, but the nomenclature references the notion of a separation profile varying smoothly between two geometrically meaningful points. The formulation enables precise, continuous anatomic assessment relevant for surgical and rehabilitation planning.

7. Generalizations, Limitations, and Open Problems

The diastasis function is a central invariant in Kähler geometry but is subject to real-analyticity constraints: its global definition typically requires real-analytic metrics, although in special symmetric or toric settings, explicit global expressions are possible (Zelditch, 2016). For merely smooth ($\varphi$7) settings, the function can only be extended in an almost-analytic sense near the diagonal, with no canonical or global geometric meaning away from it.

Calabi’s original program of classifying all Kähler immersions relies crucially on the detailed properties of the diastasis function and its expansions. While major advances have been made for homogeneous and symmetric spaces, the local-to-global extension for generic Kähler manifolds and the interplay with the (higher) Bergman kernel asymptotics remain important open problems (Loi et al., 2017). Furthermore, the extension and applicability of diastasis-based measures in high-dimensional conformal or coupling spaces, as well as in clinical or data-driven anatomical modeling, are active areas of continued research and development.

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References:

• (Zelditch, 2016, Loi et al., 2017, Mossa, 2012, Dong et al., 2021, Bachas et al., 2013, D'Hoker et al., 2014, Goto et al., 2018, Goto et al., 2020, Loi et al., 2016, Berceanu, 2013, Loi et al., 2009, Ray, 2024, Gueroult et al., 20 Jan 2025)