Orthotoric Kähler Orbifold Surfaces

• Orthotoric Kähler orbifold surfaces are compact 2-dimensional complex spaces with toric symmetry defined by a separable Hamiltonian structure.
• They employ an explicit metric ansatz using quartic polynomials and a labeled quadrilateral moment map to encode orbifold singularities.
• These models support studies in extremal, Einstein, and hyperkähler geometries and provide explicit solutions to complex pluriclosed PDEs.

Orthotoric Kähler orbifold surfaces are compact complex 2-dimensional Kähler orbifolds whose local geometry arises from a toric symmetry and can be described explicitly by the orthotoric ansatz. These metrics constitute an important subclass of toric Kähler metrics distinguished by their separable Hamiltonian structures, and they play a central role in recent advances in the study of special Kähler geometries, including pluriclosed Hermitian manifolds, extremal metrics, and their applications in complex geometry and mathematical physics.

1. Orthotoric Geometry and Toric Orbifolds

Orthotoric Kähler geometry is characterized by the existence of commuting Hamiltonian~S¹-actions giving rise to a dense open torus orbit, with the metrics determined by functions of a single variable in adapted coordinates. In the orbifold category, one considers compactifications over a generic quadrilateral moment polytope, with orbifold singularities encoded at the facets. The toric data is specified by the polytope $\Delta \subset \mathbb{R}^2$ and a set of integer weights labeling the polytope's edges, which dictate the local isotropy groups of the holomorphic torus action. The topological and analytic structure of these orbifolds generalizes the theory of Delzant polytopes to the rational-fan framework appropriate for orbifold singularities (Apostolov et al., 8 Jan 2026, Jelonek, 2021).

2. Explicit Metric Ansatz and Local Structure

In dimension four, orthotoric Kähler metrics can be written in local coordinates $(x, y, t_1, t_2)$ as: $$g = (x - y)\left(\frac{dx^2}{A(x)} + \frac{dy^2}{B(y)}\right) + \frac{1}{x-y}\left( A(x) (dt_1 + y dt_2)^2 + B(y) (dt_1 + x dt_2)^2 \right)$$ with associated Kähler form: $$\omega = dx \wedge (dt_1 + y dt_2) + dy \wedge (dt_1 + x dt_2)$$ where $A(x)$ and $B(y)$ are real quartic polynomials (subject to additional reality and positivity conditions for completeness and compactness). The coordinates $t_1, t_2$ parametrize the $T^2$ action and $(x, y)$ range over a rectangle determined by the roots of $A$ and $B$.

The Hamiltonians for the torus action are $\mu_1 = x + y$, $\mu_2 = x y$, linking the orthotoric framework to symplectic toric geometry. Imposing periodicity and orbifold conditions on the coordinates yields a compact (possibly singular) Kähler orbifold (Jelonek, 2021).

3. Solvability, Boundary, and Orbifold Compactification

To obtain a global orbifold metric, $A(x)$ and $B(y)$ must each vanish at two endpoints (say, $x=\alpha_1,\alpha_2; ~y=\beta_1,\beta_2$) corresponding to facets of the moment polytope. The slopes at these points are fixed by the orbifold weights: $$A(\alpha_i) = 0, \quad A'(\alpha_i) = (-1)^{i-1} \frac{2}{r_i}, \quad r_i > 0$$ and

$$B(\beta_j) = 0, \quad B'(\beta_j) = (-1)^{j-1} \frac{2}{p_j}, \quad p_j > 0$$

for $i, j = 1,2$ with $r_i, p_j \in \mathbb{N}$. Thus, each edge of $\Delta$ is labeled by the order of the cone singularity (angle $2\pi/r_i, 2\pi/p_j$), precisely encoding the local isotropy of the corresponding $S^1$-action. The requirement that the normals with weights form an integral lattice ensures compatibility with the global orbifold structure (Jelonek, 2021, Apostolov et al., 8 Jan 2026).

The general moduli space of inequivalent orthotoric orbifold Kähler metrics—modulo affine symmetry of $(x, y)$ and the torus variables—depends on three independent real parameters, corresponding to the cross-ratios and scaling freedoms of the polynomials and the polytope.

4. Relations to 6th-Order PDEs and Momentum Map Formalism

A central application of orthotoric Kähler orbifold surfaces is the construction of explicit solutions to certain 6th-order nonlinear PDEs, which arise in the study of Bismut Ricci-flat pluriclosed Hermitian manifolds (BHE). The underlying equation is: $$\Delta \mathrm{Scal} + \frac{1}{2} \mathrm{Scal}^2 - \|\mathrm{Ric}\|^2 - \|\theta\|^2 = 0, \quad \mathrm{Scal} > 0,$$ where $\mathrm{Scal}$ is the scalar curvature, $\theta$ a closed real $(1,1)$-form, and $\mathrm{Ric}$ the Ricci form of $\omega$. For orthotoric metrics with quadratic $A, B$, explicit calculations show that the PDE reduces to an algebraic constraint on the coefficients of $A$ and $B$: $$4(a_0 + b_0)(a_2 + b_2) = (a_1 + b_1)^2 + 4\lambda^2$$ with $\lambda$ determined by the cohomological data $[\theta]$ (Apostolov et al., 8 Jan 2026).

The infinite-dimensional GIT perspective associates to this PDE a momentum map structure on the Fréchet manifold of complex structures, yielding generalized Mabuchi and Calabi functionals whose critical points are precisely the orthotoric solutions. The Futaki invariant defines an obstruction to the existence of solutions, paralleling developments in constant scalar curvature Kähler geometry.

5. Classification and Special Cases

Orthotoric Kähler orbifold metrics subsume several distinguished cases:

• Extremal and self-dual metrics: When $A$ and $B$ are degree-four polynomials parameterizing extremal metrics, further algebraic constraints recover Bach-flat or Einstein 4-orbifolds.
• Hyperkähler structures: For special relations among the coefficients, the metric is Ricci-flat and hyperkähler, corresponding to the Gibbons-Hawking multi-center solutions.
• Couzens–Gauntlett–Martelli–Sparks (CGMS) family: Imposing the constraints $a_1 + b_1 = 0, ~ a_2 + b_2 = 0$ yields solutions relevant for compactifications in AdS$_3 \times Y_7$ supergravity, where $\theta=0$ and the equation reduces to the supersymmetry condition for certain Sasaki–Einstein manifolds ($Y_7 \cong S^2 \times S^3$). The broader orthotoric family extends these by allowing $\theta \neq 0$, leading to genuinely new types of BHE 3-folds (Apostolov et al., 8 Jan 2026).

6. Topological, Singular, and Global Features

As toric orbifolds, the global topology of orthotoric surfaces is fully encoded by the labeled quadrilateral and the four orbifold weights. These include generalizations of weighted projective planes, Hirzebruch orbifolds, and more exotic quotients. Near each orbifold locus, the metric exhibits a cyclic cone singularity of prescribed order. The compatibility of the orbifold structure with the labeled polytope is governed by rational-fan (generalized Delzant) conditions (Jelonek, 2021).

Completeness and compactness of the metric are achieved by choosing the zeros and weights so that $A(x)$ and $B(y)$ are positive in the interior and vanish simply at the ends with integral slopes.

7. Context and Applications in Complex Geometry and Mathematical Physics

Orthotoric Kähler orbifold surfaces serve as key explicit models for:

• The study of generalized extremal Kähler metrics and the role of stability (GIT) in their existence (Apostolov et al., 2010).
• The construction and classification of Bismut Ricci-flat pluriclosed 3-folds via dimensional reduction, providing the first non-Kähler BHE structures on $S^3 \times S^3$ and $S^1 \times S^2 \times S^3$ not locally isometric to Samelson geometries (Apostolov et al., 8 Jan 2026).
• Exploration of special holonomy and Einstein structures, especially via the links to both Kähler and hyperkähler geometry (Jelonek, 2021).

These families also clarify the relationship between orbifold singularities, labeled polytope data, and the existence of global toric Kähler metrics, offering a framework for concrete investigation of higher-dimensional analogues and applications in both abstract geometric analysis and string-theoretic compactification scenarios.

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For further technical developments and explicit formulae, see (Apostolov et al., 8 Jan 2026) for the connection of orthotoric orbifold metrics to sixth-order pluriclosed PDEs and momentum map structures, and (Jelonek, 2021) for a general analytic and synthetic account of generalized orthotoric surfaces, their structure equations, and orbifold compactification theory. The connections with extremal metrics, Einstein 4-orbifolds, and the CGMS construction are developed in (Apostolov et al., 2010) and contextualized in modern Kähler geometry.