Singular Toric Domains: Structures & Invariants

Updated 4 October 2025

Singular toric domains are spaces defined as preimages under moment maps that include singularities, non-free torus actions, and orbifold structures, providing key examples in algebraic and symplectic geometry.
They are characterized by explicit combinatorial data from fans, polytopes, and cones, which govern their cohomological invariants, characteristic classes, and mixed Hodge structures.
These domains bridge analytic and combinatorial techniques in studying symplectic embedding problems, resolution of singularities, and invariant computations across geometry and mathematical physics.

A singular toric domain is a generalization of the classical toric domain concept, developed to incorporate singularities, non-free torus actions, and orbifold or even more exotic structures. Singular toric domains play a foundational role in algebraic geometry, symplectic geometry, and mathematical physics due to their rich interplay between combinatorial, topological, and analytic structures. Their rigorous paper illuminates the behavior of invariants, embeddings, characteristic classes, and singularities in domains with toric symmetry.

1. Definitions and Key Structures

A toric domain in dimension $2n$ is typically defined as the preimage under the standard moment map $\mu: \mathbb{C}^n \to \mathbb{R}_{\geq 0}^n$ of a region $A \subset \mathbb{R}_{\geq 0}^n$ , i.e.: $U = \mu^{-1}(A) = \{ z \in \mathbb{C}^n : \mu(z) \in A \}.$

A singular toric domain arises when the moment region $A$ either touches one or more coordinate hyperplanes, is equipped with more general quotient or orbifold structures, or is modeled on objects such as lens spaces, resulting in non-free torus actions or orbifold/quotient singularities. In four dimensions, for example, a singular toric domain may be given as a subset of a symplectic orbifold $M_n$ constructed from a cone $V_n \subset \mathbb{R}^2$ and may have boundary a lens space $L(n,1)$ with a natural orbifold point at the apex (Trejos, 27 Sep 2025).

Singular toric domains also encompass:

Quotients $X = M/G$ , with $M$ smooth and $G$ a finite or reductive group, such that the torus action descends to $X$ and introduces finite (e.g., cyclic) quotient singularities (Maxim et al., 2013).
Toric quasifolds, modeled locally as open subsets modulo countable group actions, extending toric geometry to nonrational polytopes (Prato, 2022).
Constructions arising from cones over (possibly non-simplicial or nonrational) polytopes, and those with folded or origami-type symplectic structures (Hockensmith, 2015).

2. Combinatorial and Cohomological Invariants

The topology and cohomology of singular toric domains are controlled by combinatorial data from fans, polytopes, or associated cones:

Integral Cohomology Rings: The integral cohomology of a singular toric variety (or orbifold) is described in terms of weighted Stanley–Reisner rings (wSR $[\Sigma]$ ), modded out by integrality conditions. Under certain combinatorial hypotheses on the polytope $Q$ or fan $\Sigma$ (e.g., relative primitivity of local groups at each free vertex), the cohomology is concentrated in even degrees and is torsion-free. The presentation is (Bahri et al., 2015):

$H^*(X) \cong wSR[\Sigma]/J,$

where $J$ encodes the global linear relations tied to the fan structure.

Equivariant and Generalized Cohomology: For singular toric domains associated to almost simple or divisive polytopes, GKM theory yields explicit descriptions of equivariant cohomology, $K$ -theory, and cobordism in terms of piecewise algebras over the fan. The structure is governed by combinatorial stratifications, with the global cohomology ring determined by compatibility and divisibility relations among local Euler classes (Sarkar et al., 2018).
Mixed Hodge Structures: The singular cohomology of a proper toric variety, even singular, is mixed of Hodge–Tate type. The only nonzero Hodge numbers occur along the diagonal, with explicit formulas governed by the face numbers of the associated polytope (Kim et al., 15 May 2025).

3. Characteristic Classes and Stringy Invariants

Characteristic classes for singular toric domains account for quotient and orbifold singularities:

Motivic Chern and Hirzebruch Classes: For a finite or étale quotient $T: M \to X = M/G$ (with $M$ smooth and $G$ finite or reductive), the Zariski sheaf of $p$ -forms on $X$ is $(T_*\Omega_M^p)^G$ . Characteristic classes such as the motivic Chern class and (homology) Hirzebruch class can be computed via the “descent” of differential form data from $M$ using the Cox construction and related quotient models (Maxim et al., 2013). Explicitly:

$\mathrm{mC}_y(X) = \sum_p [\tilde{\Omega}_X^p] y^p,$

where $\tilde{\Omega}_X^p$ is the Zariski sheaf of $p$ -forms.

Stringy Chern Classes: For $Q$ -Gorenstein toric varieties with log-terminal singularities, the stringy Chern class is given by a sum over the cones in the fan, weighted by combinatorial data (the normalized volumes of the associated polytopes). The independence of these classes from the choice of resolution allows invariants such as the stringy Euler number to be computed combinatorially (Batyrev et al., 2016).

Invariant	Formula or Data Source
Motivic Chern class	$\sum_p [\tilde{\Omega}_X^p] y^p$
Stringy Chern class	$\sum_{\sigma\in\Sigma} v(\sigma) [X_\sigma]$
Cohomology ring	$wSR[\Sigma]/J$

These invariants are fundamental in intersection theory, mirror symmetry, and classification of singularities.

4. Symplectic Geometry, ECH Capacities, and Embedding Problems

Many singular toric domains arise as symplectic (or orbifold-symplectic) domains with non-free or singular torus actions. Their symplectic invariants and embedding characteristics are described as follows:

ECH Capacities: When the torus action is not free (e.g., moment regions meet coordinate axes or the boundary is a lens space), one can translate the moment region so that it does not meet the axes; then, all symplectic capacities, including Embedded Contact Homology (ECH) capacities, are invariant under translation. The ECH capacities can then be computed using Hutchings's combinatorial formulas:

$c_k(A) = \inf \{ L_A(T) : T \text{ is a lattice polygon enclosing %%%%29%%%% lattice points} \},$

where $L_A$ is a norm determined by $A$ (Landry et al., 2013).

Concave Singular Toric Domains: For domains with lens space boundary $L(n,1)$ (e.g., singular balls in orbifolds), the ECH capacity sequence is given by maximizing a 2-length over concave integral lattice paths $A$ with prescribed lattice count:

$c_k(X) = \max\{ l(A) : L(A) = k \},$

where $l(A)$ is a weighted sum over edge vectors and $L(A)$ is the lattice count (Trejos, 27 Sep 2025).

Desingularization and Obstructions: The ECH capacities of singular toric domains remain applicable to their desingularizations—such as rational blowups or almost toric fibrations—yielding sharp obstructions for symplectic embeddings into, for example, the unit cotangent bundle of $S^2$ or $\mathbb{R}P^2$ (Trejos, 27 Sep 2025, Trejos, 2023). The combinatorial structure survives under desingularization.

A plausible implication is that, for a broad class of symplectic embedding problems, singular toric domains serve as an analytic-combinatorial bridge between orbifold invariants and smooth manifold embedding obstructions.

5. Resolution of Singularities and Birational Geometry

The classification and resolution of singularities in toric domains leverage the combinatorial nature of fans and polytopes:

Blow-Ups of Singularities: For three-dimensional terminal toric singularities, all purely log terminal blow-ups and canonical divisorial contractions fall into a dichotomy: either the contraction is toric (read directly from the fan and weighted blow-up data) or is non-toric, characterized by non-toric centers on the exceptional divisor (Kudryavtsev, 2014). The presence of non-toric subvarieties (not in any one-dimensional orbit) forces the contraction to be non-toric.
Classification of Surface Singularities: Two-dimensional toric log germs can be classified via minimal log discrepancy (mld) thresholds. Except for finitely many exceptions, there exist invariant hyperplane sections $H = m_1 E_1 + m_2 E_2$ such that $(X, B+tH)$ remains log canonical, with the data fully controlled by lattice linear inequalities (Ambro, 29 Jul 2024). This enables a combinatorial classification of singularities in the toric setting.

6. Applications and Extensions

Singular toric domains have impact in several domains:

Algebraic Statistics and Learning Theory: Singular learning machines with non-invertible Fisher information can be interpreted and resolved via toric algebraic geometry, associating Newton polytopes and lattice cones to error polynomials and using toric resolutions to compute learning coefficients in statistical models (Castillo-Villalba et al., 2017).
Height Zeta Functions and Arithmetic Applications: In arithmetic geometry, counting integral points on singular (simplicial, but not regular) projective toric varieties is controlled by a harmonic analysis approach wherein the main term of the height zeta function is obtained via multidimensional residue calculus. This method remains valid in the presence of cyclic quotient singularities, and, as an application, leads to precise asymptotics for the number of integral monic polynomials with prescribed cyclic Galois group (O'Desky, 1 Oct 2024).
Toric Quasifolds: Extending beyond rational polytopes, toric quasifolds model highly singular toric domains through quotienting by countable (possibly non-Hausdorff) group actions, with significance for applications in quasicrystal theory and noncommutative geometry (Prato, 2022).

7. Singular Cohomology, Lefschetz Morphisms, and Local Invariants

Singular toric domains serve as model spaces to investigate the relationship between singular cohomology, mixed Hodge modules, and local cohomological properties:

Local Cohomology and the Ishida Complex: The local cohomology and Du Bois complex of a toric variety are described in terms of the combinatorial Ishida complex, encoding depth conditions and local invariants. Vanishing or nonvanishing in the Ishida complex reflects the local cohomological defect, an important invariant not determined purely by the combinatorial data.
Lefschetz Morphisms: For a proper toric variety $X$ and a line bundle $L$ , the cup product with $c_1(L)$ on singular cohomology corresponds (via duality) to connecting homomorphisms in short exact sequences of Ishida complexes. This links global Lefschetz theorems to local information about singularities (Kim et al., 3 Jun 2025).
Non-combinatoriality of Defects: The local cohomological defect is generally not a combinatorial invariant of the fan, with explicit examples in dimension 4 where toric varieties of the same combinatorial type have different defects.

Singular toric domains, both as algebraic and symplectic objects, serve as a testing ground for the interplay of combinatorics, geometry, and analysis in singular spaces. Their paper leverages explicit fan/polytope data to describe cohomological invariants, characteristic classes, symplectic capacities, and birational operations, providing a unifying framework across algebraic, symplectic, and arithmetic geometry as well as applications in representation theory and learning theory.