A canonicity criterion for toric varieties and the classification of canonical 4-simplices

1. Problem Statement

The paper addresses two interconnected problems in algebraic geometry and combinatorics:

1. Characterization of Singularities: Establishing a practical criterion to determine when a toric variety is canonical or terminal. These are specific types of mild singularities crucial in the Minimal Model Program (MMP) and the classification of Fano varieties.
2. Classification of Fake Weighted Projective Spaces (FWPS): Specifically, the paper aims to classify all canonical and terminal fake weighted projective spaces in dimension four. FWPS are $\mathbb{Q}$-factorial projective toric varieties of Picard number one. They generalize weighted projective spaces and correspond to lattice simplices with the origin as their only interior lattice point (canonical polytopes).

While classifications exist for dimensions 2 and 3, and for terminal cases in dimension 4, a complete classification of canonical 4-dimensional FWPS was previously unknown.

2. Methodology

The author develops a novel approach based on the Reid–Shepherd-Barron–Tai (RSBT) criterion for quotient singularities, adapted to the language of toric geometry and local class groups.

A. Theoretical Framework

• Cox's Quotient Construction: The toric variety $X$ is viewed as a quotient $\hat{X}/H$, where $\hat{X} \subseteq \mathbb{C}^r$ is an open toric subvariety and $H$ is a quasitorus acting diagonally.
• Local Class Groups: For a toric fixed point $x_i \in X$, the local structure is determined by a finite subgroup $H_i \subseteq H$. The character group of $H_i$ is isomorphic to the local class group $\text{Cl}(X, x_i)$.
• The "Age" Criterion: The RSBT criterion states that a quotient singularity is canonical (resp. terminal) if and only if the age of every non-identity element in the acting group is $\ge 1$ (resp. $> 1$).
  • The age of an element $h$ is the sum of the fractional parts of the exponents of its eigenvalues acting on the tangent space.
• Main Theorem (Theorem 1.1 & 2.7): The paper proves that a $\mathbb{Q}$-factorial projective toric variety $X$ is canonical if and only if $\text{age}(h) \ge 1$ for all $1 \neq h \in H_i$ (for all fixed points). This translates the geometric condition into a combinatorial condition on the degree matrix $Q$ associated with the fan.

B. Algorithmic Implementation

For the specific case of FWPS (Picard number one), the author derives an integer-based criterion (Theorem 3.2) and an iterative classification algorithm (Algorithm 3.10):

1. Weight Vectors: Start with a list of valid weight vectors $w = (w_0, \dots, w_n)$ that define canonical weighted projective spaces (using known bounds).
2. Torsion Vectors: Iteratively adjoin "torsion rows" (representing the finite part of the class group) to the degree matrix.
3. Minimality and Equivalence: To avoid counting isomorphic varieties multiple times, the algorithm enforces row-minimality and equivalence checks (Definition 3.5 & 3.7) based on the automorphism group of the underlying abelian group.
4. Filtering: At each step, only extensions that preserve the canonicity/terminality condition (via the age sum formula) are retained.

C. Fine Interior Analysis

After generating the list of simplices, the paper analyzes their Fine interiors ($F(\Delta)$).

• The Fine interior is a rational polytope defined by Batyrev, which determines the Kodaira dimension of the associated Calabi–Yau hypersurface.
• The author computes $F(\Delta)$ for all generated 4-simplices to determine the dimension of the interior and the stringy Euler number of the resulting Calabi–Yau models.

3. Key Contributions

1. New Canonicity Criterion: A rigorous characterization of canonicity and terminality for simplicial toric varieties in terms of local class group actions and the "age" of group elements (Theorem 2.7).
2. Classification Algorithm: A general, efficient algorithm (Algorithm 3.10) for classifying canonical and terminal FWPS in any dimension $n$.
3. Dimension 4 Classification: The first complete enumeration of canonical fake weighted projective spaces in dimension four.
4. Fine Interior Distribution: A comprehensive statistical analysis of the Fine interiors of these 4-simplices, including the distribution of their dimensions and the calculation of stringy Euler numbers.

4. Results

A. Classification of 4D Canonical FWPS

• Total Count: There are 710,450 non-isomorphic canonical fake weighted projective spaces of dimension four.
• Comparison:
  • The algorithm reproduces Kasprzyk's known results for dimension 3 (225 canonical, 35,947 terminal).
  • For dimension 4, it confirms the 35,947 terminal spaces and adds the new count of 710,450 canonical spaces.
• Performance: The algorithm runs on a midrange computer (16 threads) and completes in approximately 12 minutes.
• Data Availability: The defining data (degree matrices) for all 710,450 polytopes are made available in a repository (cited as [20]).

B. Fine Interior Statistics

The paper analyzes the Fine interiors ($F(\Delta)$) of the 710,450 canonical 4-simplices:

Dimension of $F(\Delta)$	Number of Simplices
0 (Calabi–Yau candidates)	387,310
1	112,672
2	95,713
3	70,130
4	44,625

• Observation: Unlike in dimension 3 (where the zero-dimensional case is overwhelmingly dominant), the zero-dimensional case in dimension 4 is less dominant, and all dimensions from 0 to 4 occur.

C. Stringy Euler Numbers

For the 387,310 simplices with a zero-dimensional Fine interior (which correspond to Calabi–Yau hypersurfaces):

• The stringy Euler number takes 94,233 distinct values.
• Only 852 of these values are integers.
• The paper provides the frequency distribution of these values (Figure 1 in the original text).

D. Distinct Fine Interiors

The paper notes that while there are 710,450 simplices, they map to far fewer distinct Fine interiors up to unimodular equivalence:

• Dim 1: 46 distinct interiors.
• Dim 2: 2,178 distinct interiors.
• Dim 3: 7,825 distinct interiors.
• Dim 4: 16,072 distinct interiors.

5. Significance

• Mirror Symmetry: Canonical polytopes serve as candidate Newton polytopes for Calabi–Yau hypersurfaces in mirror symmetry. This classification significantly expands the known database of such polytopes in dimension 4, providing new candidates for constructing mirror pairs.
• Birational Geometry: The analysis of Fine interiors provides deep insights into the birational geometry of toric hypersurfaces. The dimension of the Fine interior directly dictates the Kodaira dimension of the canonical model.
• Computational Algebraic Geometry: The paper demonstrates the feasibility of classifying high-dimensional toric varieties using group-theoretic criteria and efficient sieving algorithms, bridging the gap between theoretical criteria (RSBT) and computational enumeration.
• Calabi–Yau Moduli: By computing stringy Euler numbers for the 0-dimensional Fine interior cases, the paper contributes to the understanding of the moduli space of Calabi–Yau threefolds (since a 4-simplex defines a 3-fold hypersurface).

In summary, Ghirlanda provides a powerful theoretical tool for checking singularities and executes a massive computational classification that fills a critical gap in the literature regarding 4-dimensional canonical toric varieties and their associated Calabi–Yau geometries.