Noncommutative Quasi-Crepant Resolution

• Noncommutative quasi-crepant resolutions are categorical constructions that replace classical crepant resolutions using tilting theory and combinatorial geometry.
• They employ (partial) tilting complexes on smooth toric DM stacks from regular triangulations of lattice polytopes to yield endomorphism algebras with finite global dimension.
• This approach generalizes established NCCR methods by integrating tilting bundles, reflexive polytopes, and derived category frameworks with applications in mirror symmetry.

A noncommutative quasi-crepant resolution is a categorical or algebraic replacement for a classical (commutative) crepant resolution in algebraic geometry, designed to address the resolution of singularities in situations where a traditional resolution may not exist or is difficult to describe. For affine toric Gorenstein varieties, recent developments have brought together techniques from tilting theory, derived categories, Deligne–Mumford stacks, and combinatorial geometry to construct and analyze such noncommutative resolutions. These approaches focus on finding endomorphism algebras of (partial) tilting objects on smooth toric stacks associated to regular triangulations of lattice polytopes, thereby yielding noncommutative crepant resolutions (NCCRs) and their generalizations.

1. Unified Framework and Main Generalizations

The central advancement is a common generalization of the NCCR constructions of Špenko–Van den Bergh and of Iyama–Wemyss, which allows for the systematic production of NCCRs for a broad class of affine toric Gorenstein varieties (Malter et al., 15 Sep 2025). On one side, Špenko–Van den Bergh's method realizes NCCRs as endomorphism algebras of tilting bundles over smooth toric Deligne–Mumford (DM) stacks, constructed from regular (crepant) triangulations of a given lattice polytope. On the other, Iyama–Wemyss established that sufficiently ample tilting complexes on a projective birational model of a Gorenstein singularity yield NCCRs via their endomorphism rings.

The innovation in this new framework is threefold:

• It incorporates not only tilting bundles but also partial tilting complexes, thus handling cases where a full tilting bundle may not exist.
• The method works with crepant morphisms from smooth DM stacks associated to simplicial fans derived from regular triangulations, independently of the existence of smooth projective (commutative) resolutions.
• The construction is functorially connected to the combinatorics of the underlying polytope P corresponding to the toric Gorenstein variety.

2. Construction Methodology: Toric DM Stacks and Tilting Theory

Given a lattice polytope $P \subset \mathbb{R}^n$ with nonempty interior, the corresponding Gorenstein cone is $\sigma = \mathrm{Cone}(P \times \{1\})$, and the coordinate ring of the associated affine toric Gorenstein variety is $R = k[\sigma^\vee \cap M]$, where $M$ is the dual lattice. A regular triangulation of $P$ produces a simplicial fan $\Sigma$, which yields a smooth toric DM stack $\mathcal{X}_\Sigma$ (the Cox stack).

A (partial) tilting complex $\mathcal{T}$ is constructed on $\mathcal{X}_\Sigma$ so that its endomorphism algebra

$$\Lambda = \mathrm{End}_{\mathcal{X}_\Sigma}(\mathcal{T})$$

has finite global dimension and is Cohen–Macaulay as an $R$-module. The key is that $\Lambda$ can then be expressed as $\mathrm{End}_R(M)$ for some reflexive $R$-module $M$. The finite global dimension implies homological smoothness, while the Cohen–Macaulay property over $R$ captures the crepancy of the resolution.

The main theoretical statement is: if $\mathcal{T}$ is a (partial) tilting complex on $\mathcal{X}_\Sigma$ with $\mathrm{gl.dim}(\Lambda)<\infty$, then $\Lambda$ is an NCCR of $R=k[\sigma^\vee \cap M]$.

In summary, the process is:

1. Choose a lattice polytope $P$ with interior point guaranteeing Gorensteinness.
2. Triangulate $P$ regularly to obtain a simplicial fan.
3. Construct the stack $\mathcal{X}_\Sigma$ via the Cox construction.
4. Build a (partial) tilting complex $\mathcal{T}$ on $\mathcal{X}_\Sigma$.
5. Form $$\Lambda = \mathrm{End}_{\mathcal{X}_\Sigma}(\mathcal{T})$$ and verify its properties as an NCCR for $R$.

3. The Role of Polytopes and Reflexivity

The polytope $P$ encodes both the singularity type and the possible resolutions.

• The condition that $P$ has an interior lattice point ensures that the associated cone is Gorenstein, which is a precondition for crepancy.
• Reflexivity of the polytope ($P^\vee$ is also a lattice polytope), which plays a leading role in mirror symmetry and Fano geometry, is crucial for constructing explicit tilting bundles via exceptional collections.
• Regular triangulations of $P$ provide the combinatorial data necessary for the DM stack to be smooth and for the existence of acyclic line bundles used in tilting objects.

When $P$ is reflexive with at most $(\dim P + 2)$ vertices, the existence of full strong exceptional collections—and hence tilting bundles—can be established using results of Borisov and Hua, ensuring that the endomorphism algebra yields an NCCR (Malter et al., 15 Sep 2025).

4. Special Cases: Reflexive Polytopes with Few Vertices

For reflexive polytopes with $\leq \dim P + 2$ vertices, the Borisov–Hua theorem guarantees the presence of a tilting bundle on the corresponding smooth toric DM stack. The forbidden cone techniques allow the explicit construction of exceptional collections, ensuring the vanishing of higher cohomology groups for pairs of line bundles—precisely the acyclicity required for tilting.

The upshot: for such $P$, the described procedure always produces an NCCR of $R$ by forming the endomorphism algebra of this tilting bundle. The necessary vanishing

$$H^i(\mathcal{X}_\Sigma, \mathcal{T}^\vee \otimes \mathcal{T} \otimes (\omega^*_{\mathcal{X}_\Sigma})^{\otimes l}) = 0 \quad \text{for} \quad i \neq 0, l > 0$$

follows from the forbidden cone and acyclicity properties in the Borisov–Hua framework.

5. Key Mathematical Formulations and Criteria

Let $R = k[\sigma^\vee \cap M]$ be the coordinate ring of an affine toric Gorenstein variety, with $\sigma=\mathrm{Cone}(P \times \{1\})$ for $P\subset \mathbb{R}^n$ a lattice polytope containing the origin in its interior. A regular triangulation of $P$ yields a Cox stack $\mathcal{X}_\Sigma$. A (partial) tilting complex $\mathcal{T}$ on $\mathcal{X}_\Sigma$ produces an algebra

$$\Lambda = \mathrm{End}_{\mathcal{X}_\Sigma}(\mathcal{T})$$

having the properties:

• $\mathrm{gl.dim}(\Lambda)<\infty$ and
• $\Lambda$ is Cohen–Macaulay as an $R$-module (i.e., $\Lambda\in\mathrm{CM}\,R$), hence $\Lambda \cong \mathrm{End}_R(M)$ for reflexive $M$, and is an NCCR of $R$.

In the reflexive polytope case (especially with $\leq \dim P+2$ vertices), tilting bundles can be explicitly constructed using exceptional line bundles associated to the forbidden cones, as in the Borisov–Hua setup.

6. Implications, Applications, and Connections

This theory yields significant advances:

• It enlarges the collection of affine Gorenstein toric singularities known to admit explicit noncommutative crepant resolutions, going beyond the existence of commutative (geometric) crepant resolutions.
• The methodology is robust under variation of GIT quotients and extends to settings where only DM stacks (not smooth varieties) are available as ambient spaces.
• Connections to mirror symmetry, especially for reflexive polytopes, are reinforced by the role of toric Fano and almost Fano varieties.
• The flexibility to use partial tilting complexes broadens the applicability to singularities where full tilting bundles may not exist.
• Derived categories of these NCCRs are expected to govern birational geometry through Bondal–Orlov–Kawamata-type philosophies, i.e., all such resolutions are conjectured to be derived equivalent when they exist.

The construction also opens the way for concrete computations and further generalizations, including to non-toric or mildly singular ambient spaces by similar stack-theoretic and categorical methods.

7. Future Directions and Open Questions

Further research directions prompted by this work include:

• Classification of all toric Gorenstein singularities admitting NCCRs via this methodology, and the characterization of necessary conditions on $P$ (e.g., reflexivity, triangulability).
• Extension of these categorical techniques to other classes of singularities, especially beyond the toric or Gorenstein framework, possibly via stacky or derived enhancements.
• Deeper exploration of mirror symmetry predictions, leveraging the polytope combinatorics and derived invariants from NCCRs.
• Exploration of the derived equivalence class of all such NCCRs (in line with the noncommutative Bondal–Orlov conjecture), particularly how mutations or wall crossings in GIT affect the associated tilting objects and derived categories.
• Investigation of the interaction of these NCCRs with Frobenius and D-module techniques in positive and mixed characteristic.

In conclusion, the synthesis of tilting theory, toric stack geometry, and polytope combinatorics provides a systematic and explicit pathway to constructing noncommutative quasi-crepant resolutions for a large and intricate class of affine toric Gorenstein varieties, advancing both the birational and categorical understanding of resolutions in algebraic geometry (Malter et al., 15 Sep 2025).