Kähler Toric Manifolds

• Kähler toric manifolds are complex 2n-dimensional spaces equipped with a Kähler structure and an effective Hamiltonian torus action, linking symplectic geometry with convex polytopes.
• Their structure is encoded by strictly convex symplectic potentials that yield invariant Kähler metrics, where the Delzant polytope uniquely determines the manifold’s geometry.
• Advanced tools like the Abreu equation, Legendre transforms, and stability conditions underpin their analysis, impacting studies in mirror symmetry and generalized Kähler theories.

A Kähler toric manifold is a complex manifold of real dimension $2n$ equipped with a Kähler structure and an effective, holomorphic, Hamiltonian action of an $n$-dimensional real torus. The geometry of such manifolds is governed by the interplay between symplectic geometry, combinatorial convexity via moment map images, and the differential geometry of invariant Kähler metrics encoded by symplectic potentials. The theory has deep connections to combinatorics, convex analysis, PDEs of Hessian and Monge–Ampère type, stability in algebraic geometry, and modern developments in generalized complex geometry and mirror symmetry.

1. Foundational Structure: Symplectic and Complex Toric Geometry

A Kähler toric manifold is a quadruple $(M^{2n}, \omega, T^n, \mu)$, where:

• $M$ is a compact connected $2n$-manifold,
• $\omega$ is a symplectic form on $M$,
• $T^n \cong (S^1)^n$ acts effectively by symplectomorphisms,
• $$\mu: M \rightarrow \mathfrak{t}^* \cong \mathbb{R}^n$$ is a moment map satisfying $$\iota_{X_{\xi}} \omega = d \langle \mu, \xi \rangle$$ for all $\xi \in \mathfrak{t}$.

The image $\Delta = \mu(M)$ is a convex polytope of the form $$\{x \in \mathfrak{t}^* : \langle v_j, x \rangle + \lambda_j \geq 0, j=1,\dots,d\}$$, with primitive integral normals $v_j$ at each facet and the Delzant condition ensuring that at each vertex, the $n$ normals meeting there form a $\mathbb{Z}$-basis. Delzant’s theorem guarantees that every such polytope arises from a unique (up to equivariant symplectomorphism) compact toric symplectic manifold, which can be constructed by symplectic reduction of $\mathbb{C}^d$ at the level defined by the $\lambda_j$ (Boulanger, 2015, Apostolov, 2022, Arezzo et al., 2014).

On the open dense orbit $M^\circ \simeq \Delta^\circ \times T^n$, action–angle (Darboux) coordinates $(x^i, \theta^i)$ provide a local model with $\omega = \sum_i dx^i \wedge d\theta^i$. The complex structure is encoded by a strictly convex symplectic potential, leading to a complete equivalence between the data of the polytope and the geometry of the manifold.

2. Toric Kähler Metrics and Symplectic Potentials

Invariant Kähler metrics $(\omega, J)$ compatible with the torus action are in bijection with strictly convex symplectic potentials $\tau \in C^\infty(\Delta^\circ)$ satisfying Guillemin boundary conditions:

$$\tau(x) = \frac{1}{2}\sum_{j=1}^d \ell_j(x) \log \ell_j(x) + \text{smooth},$$

where $\ell_j(x) = \langle v_j, x \rangle + \lambda_j$.

The Kähler metric in action–angle coordinates takes the form: $$g = \sum_{i,j} \tau_{ij} dx^i dx^j + \sum_{i,j} \tau^{ij} d\theta^i d\theta^j,$$ where $$\tau_{ij} = \frac{\partial^2 \tau}{\partial x^i \partial x^j}$$ and $(\tau^{ij}) = (\tau_{ij})^{-1}$ (Boulanger, 2015, Apostolov, 2022, Arezzo et al., 2014).

The Legendre transform gives holomorphic coordinates $z_j = e^{y_j + i\theta_j}$, $y_j = \partial_{x_j} \tau(x)$, in which the Kähler potential is $\phi(y) = \langle x, y \rangle - \tau(x)$. The positivity of the Hessian ensures the Kähler condition, while the explicit boundary behavior extends the metric smoothly to $M$.

3. Scalar Curvature, Extremal Metrics, and the Abreu Equation

The scalar curvature of a toric Kähler metric is given by Abreu’s formula: $$S(x) = -\sum_{i,j = 1}^n \frac{\partial^2 \tau^{ij}}{\partial x^i \partial x^j}.$$ The extremal Kähler condition corresponds to $S(x)$ being affine-linear on $\Delta$. Constant scalar curvature Kähler (cscK) metrics are those for which $S(x)$ is constant.

The Calabi functional,

$$\mathrm{Cal}(g) = \int_M (S - \bar S)^2 \, \omega^n,$$

is minimized by extremal metrics. The existence of cscK and extremal toric Kähler metrics is intimately tied to combinatorial stability conditions of the polytope, such as uniform K-stability, expressed in terms of the (Donaldson–)Futaki invariant for convex test functions $f$ on $\Delta$ (Apostolov, 2022, Liu, 2020). Properness of the K-energy functional is equivalent to uniform K-stability, and no extremal metric can exist unless the relevant stability is satisfied (Liu, 2020).

4. Generalizations: Generalized Kähler and Beyond

Generalized Kähler structures of symplectic type extend the classical theory by introducing a pair of commuting generalized complex structures $(\mathcal{J}_1, \mathcal{J}_2)$, with $\mathcal{J}_2$ being a $B$-transform of the standard symplectic generalized complex structure. Such structures on toric manifolds are characterized by a pair $(\tau(x), C)$, where $C$ is an antisymmetric constant matrix. The ordinary Kähler case corresponds to $C = 0$, while nontrivial $C$ realizes unobstructed holomorphic Poisson deformations in the sense of Goto (Boulanger, 2015, Wang, 2018).

The resulting bi-Hermitian or bi-complex geometry admits a strong Hamiltonian torus action, and the average of $J_+$ and $J_-$ recovers the ordinary toric Kähler structure. The moment map and associated stability story generalize, introducing new invariants such as the generalized Hermitian scalar curvature.

Generalizations further encompass non-compact toric manifolds, weighted scalar curvature and extremal metrics (including $f$-extremal and conformally Kähler, Einstein–Maxwell metrics), and the construction of scalar-flat Kähler metrics with mixed-type or conical ends via the Donaldson ansatz (Feng, 2024).

5. Non-Compact and Degenerate Toric Kähler Structures

The theory extends to certain non-compact toric symplectic manifolds, determined by polytopes with “strictly unbounded” directions. For these, scalar-flat Kähler metrics are constructed via convex symplectic potentials $u(x)$ on unbounded polytopes, with explicit boundary asymptotics corresponding to the underlying geometry of the toric divisors and ends (Poincaré, ALE, or Taub–NUT type). The metric is encoded entirely by the solution of the scalar-flat equation

$$\sum_{i,j} \frac{\partial^2 u^{ij}}{\partial x_i \partial x_j} = 0,$$

with boundary terms dictated by the combinatorics of the polytope (Feng, 2024).

In the setting of general Lagrangian torus fibrations with elliptic singularities, the Abreu–Guillemin formalism is subsumed in a coordinate-free correspondence between invariant Kähler metrics and pairs consisting of an elliptic connection on the fibration and a “hybrid b-metric” on the Delzant subspace of the integral affine base (Fernandes et al., 2024). Extremal metrics correspond to those with scalar curvature affine on the base, generalizing the affine PDE and residue structure.

6. Connections to Hessian and Dually Flat Geometry

There is a bijective correspondence between dually flat manifolds $(M, h, \nabla)$ and (real-analytic, regular, equivariant) Kähler toric manifolds. In this correspondence, the Hessian metric on $M$ is inherited from the symplectic potential of the toric Kähler manifold; the toric structure manifests as a principal $T^n$-bundle over the base of the moment map, and the Legendre transform links the respective affine coordinate descriptions. Affine isometries lift to equivariant Kähler immersions, and classical maps such as the Veronese and Segre embeddings are interpreted as lifts of inclusion maps between statistical manifolds. This unifies the geometric and probabilistic structures that underpin, for example, geometric quantum mechanics (Molitor, 2021).

7. Applications and Examples

Classical examples include:

• Complex projective space $\mathbb{C}P^n$ with the Fubini–Study metric, corresponding to the standard simplex as Delzant polytope.
• Product manifolds, with decoupled symplectic potentials.
• Hirzebruch surfaces and their equivariant blow-ups, with associated trapezoidal or cut polytopes.
• Non-compact examples such as the complement of toric divisors in strictly unbounded toric surfaces, supporting complete scalar-flat Kähler metrics with precisely specified end-behavior (Feng, 2024).
• Two-parameter toric manifolds (e.g., $\mathbb{P}^1\times\mathbb{P}^1$, $\mathbb{F}_3$, Calabi-Yau in weighted projective spaces), with their quantum cohomology and mirror symmetry structures completely determined by localization and intersection computations on moduli spaces of toric polynomial maps (Jinzenji, 2010).

The combinatorial and analytic structure of Kähler toric manifolds permeates constructions in symplectic and algebraic geometry, generalized Kähler theory, and mirror symmetry, providing a testbed for deep problems at the intersection of complex, symplectic, and algebraic geometry.