Geometric McKay Correspondence

• Geometric McKay correspondence is a framework that unifies finite group representations, quiver combinatorics, and algebraic geometry to construct singularity resolutions via module-theoretic methods.
• It employs large and almost large modules from quiver algebras, using matrix-valued evaluations and symplectic reduction to parametrize exceptional loci in resolutions.
• The method generalizes classical ADE bijections to non-isolated, toric, and non-Gorenstein cases, offering a unified toolkit for both commutative and noncommutative geometry.

The geometric McKay correspondence is a framework connecting the representation theory of finite groups, the combinatorics of quivers, and the geometry of resolutions of surface singularities. Its modern variants incorporate noncommutative algebra, quiver representations, symplectic reductions, and module-theoretic techniques to generalize and extend the classical bijections between group representations and exceptional divisors in minimal resolutions. The correspondence now encompasses the settings of noncommutative singularities, toric and non-isolated cases, and provides a toolkit for constructing both commutative and noncommutative resolutions, bridging and extending representation theory and algebraic geometry (Beil, 2011).

1. Conjectural Method via Matrix-Valued Function Rings

The geometric construction begins with noncommutative algebras arising as path algebras of quivers with relations, notably square superpotential or preprojective algebras. Given such an algebra $A$ (module-finite over its center $Z$), one constructs its "impression" $T\colon A \hookrightarrow \mathrm{End}_B(B^d)$, where $B$ is a commutative coordinate ring and $B^d$ is a free module of rank $d$. Evaluation at maximal ideals of $B$ yields matrix-valued functions on $T$.

The key objects in this approach are:

• Large modules: irreducible $A$-modules of maximal possible dimension vector. They correspond bijectively to points of the smooth locus $\operatorname{Max} Z$.
• Almost large modules: modules that are "as large as possible" except for the vanishing of a minimal path-like set $P$, encoded by a maximal chain of annihilators $0 = P_0 \subset P_1 \subset \ldots \subset P_k = P$.

Parametrizing almost large modules yields spaces isomorphic (as schemes) to exceptional loci in (commutative) resolutions of singularities. Symplectic reduction in the representation space realizes these as families parameterized by, e.g., projective spaces (such as $\mathbb{P}^1$), shrinking to points in the noncommutative spectrum as certain parameters degenerate.

This technology generalizes the classic association of irreducible representations of a finite subgroup of $GL(n, \mathbb{C})$ to exceptional divisors on the resolution of $\mathbb{C}^n / G$, with the McKay quiver encoding the relevant representation-theoretic data.

2. Verification in Archetypal Examples

The validity of the conjectural method is demonstrated in several settings:

Singularity Type	Quiver/Algebra	Exceptional Locus Parametrization
Cyclic quotient surface (type $\frac{1}{r}(1, b)$)	McKay quiver	Maximal chains of $P$-annihilators correspond to divisors in the Hirzebruch–Jung resolution
Kleinian $D_n, E_6$ singularities	Preprojective	Chains of almost large modules correspond to irreducible components of exceptional loci in minimal resolutions $Y \to \mathbb{C}^2/G$
Conifold (non-isolated threefold singularity)	Superpotential	Two families of almost large modules parameterize the two small resolutions; coordinates match those from the standard blowup
Non-isolated $3$-fold quotient ($\mathbb{C}^3/G$ abelian)	Quiver / Toric	Families of almost large modules stratified and parameterized by del Pezzo surfaces and points, agreeing with the toric combinatorics

In each case, the impression $T$ provides coordinates; maximal chains of $P$-annihilators define families of almost large modules, the parameter spaces of which manifest as the exceptional loci (curves, surfaces, or higher strata) in the classical geometric resolution.

3. Noncommutative Resolutions and Symplectic Techniques

Noncommutative resolutions, in contrast to classical (commutative) ones, replace the singular coordinate ring $R$ by a module-finite noncommutative algebra $A$ with $Z \cong R$. The geometry is captured in the representation theory of $A$, with the following features:

• The moduli space of large $A$-modules recovers the smooth locus, while stratifications by almost large modules encode exceptional loci.
• The impression $T$ embeds $A$ into a matrix algebra, allowing "commutative" geometry through the points of $\operatorname{Max} B$.
• Symplectic reduction is implemented by considering moment maps $H(z_1, ..., z_n) = (|z_1|^2 + ... + |z_n|^2)/2$, performing reductions analogous to GIT quotients, resulting in shrinking families of modules parameterized by projective spaces to "ramified point-like spheres".

This method realizes a bridge between noncommutative algebra (module categories and quivers) and classical geometry (exceptional loci and their moduli).

4. Shrinking of Exceptional Loci and Ramified Spheres

The geometric effect of noncommutative resolution is the "shrinking" of families parametering exceptional loci to point-like objects in the module category:

• Almost large modules have a shared one-dimensional socle corresponding to a vertex simple module. Under symplectic reduction, the whole family parameterizing an irreducible component collapses to a single non-Azumaya point (the annihilator of the vertex simple).
• For cyclic, Kleinian, and conifold examples, this process precisely recovers the intersection combinatorics of the exceptional divisors, as encoded by maximal chains of $P$-annihilators. For instance, in the conifold, one $\mathbb{P}^1$ shrinks to $S_1$, another to $S_2$.

This "ramified, point-like sphere" concept provides a new geometric intuition: noncommutative geometry captures both global geometry and local "singular" behavior within the same unified representation-theoretic approach.

5. Generalization and Impact on Algebraic Geometry

The presented framework substantially generalizes the classical McKay correspondence, with these salient advances:

• The mechanism applies to arbitrary finite subgroups (including those not contained in $SL_2(\mathbb{C})$), non-isolated singularities, and non-toric or non-Gorenstein cases.
• There is a canonical dictionary: large modules $\leftrightarrow$ smooth points; parameterized families of almost large modules (projective spaces arising from maximal chains) $\leftrightarrow$ exceptional divisors/components.
• The combinatorics of chains in the module category and quiver structure matches, via impression homomorphisms, the expected geometry of the exceptional locus, extending to situations where classical resolution techniques are less effective.

Potential impacts include:

• New constructions of (commutative and noncommutative) resolutions via module and quiver categories.
• Unified language bridging representation theory, GIT/symplectic reduction, and algebraic geometry.
• Extensions to higher-dimensional, non-toric, or non-Gorenstein singularities, and potential new tools for constructing derived equivalences and understanding singularities beyond reach of current methods.

6. Synthesis

The geometric McKay correspondence, as reformulated here, consists of extracting (commutative or noncommutative) resolutions of surface and threefold singularities from the representation theory and module categories of suitably chosen quiver algebras. Large and almost large modules encode the global and exceptional geometry, connecting with both classical ADE-type results and extending into new domains via symplectic/GIT techniques. This yields a powerful generalization synthesizing noncommutative algebra, quiver combinatorics, and geometric invariant theory, expanding and deepening the scope of the McKay correspondence (Beil, 2011).