Ishii's conjecture and Bridgeland stability conditions for dihedral reflection groups

Abstract: We provide a new proof of Ishii's conjecture for any dihedral reflection group $G\subset GL_2(\mathbb{C})$ from the viewpoint of Bridgeland stability conditions. Our strategy is to reduce the problem, via the derived McKay correspondence, to a geometric construction of Bridgeland stability conditions on the root stack of the maximal resolution along the strict transform of the discriminant divisor.

• The paper proves Ishii’s conjecture for dihedral reflection groups by leveraging Bridgeland stability conditions and derived category techniques, linking moduli spaces with the birational geometry of surface quotient singularities.
• It employs the derived McKay correspondence and wall-crossing analysis to directly relate GIT stability parameters with Bridgeland moduli, ensuring a precise description of the stability manifold.
• The study’s methods offer a framework that may generalize to other finite groups and higher-dimensional singularities, potentially advancing research in the minimal model program and mirror symmetry.

Ishii's Conjecture and Bridgeland Stability for Dihedral Reflection Groups

Introduction and Theoretical Context

The paper provides a new proof of Ishii’s conjecture for dihedral reflection groups in $\mathrm{GL}_2(\mathbb{C})$ by leveraging Bridgeland stability conditions, derived categories, and the McKay correspondence. The main focus is on relating birational geometry of surface quotient singularities to moduli spaces of stability conditions, extending the geometric realization of resolutions of the quotient $C^2/G$ as moduli spaces, from the context of crepant resolutions to more general (not necessarily crepant) cases described by Ishii’s conjecture.

The original conjecture of Craw and Ishii, in dimension three, addresses the realization of all crepant resolutions of $C^3/G$ (for $G \subseteq \mathrm{SL}_3(\mathbb{C})$) as moduli spaces of $\theta$-stable representations for varying King stability conditions $\theta \in \Theta(G)$. This conjecture has been proven in multiple generalities. Ishii’s generalization for the surface ($n=2$) case conjectures that every resolution of $C^2/G$ dominated by the maximal resolution arises as a moduli space of $0$-stable $G$-constellations for some generic King stability parameter $C^2/G$0.

Previous results establish the conjecture for abelian $C^2/G$1 and for particular cases; the "if" direction generally, and for small subgroups. Capellan, using GIT methods, established the result for dihedral reflection groups. This paper offers a fundamentally different route, providing a derived and categorical perspective via Bridgeland stability and wall-crossing.

Methodology: Derived Categories and Stability Manifolds

The central strategy involves the reduction of the moduli problem for $C^2/G$2 via the derived McKay correspondence, together with geometric constructions of Bridgeland stability on stack-theoretic models associated to resolutions. The core constructions are as follows:

• The use of the bounded derived category $C^2/G$3 for coherent sheaves with compact support on stacks $C^2/G$4 associated to the quotient and its resolution.
• Identification of the relevant DM stack $C^2/G$5, modeled as a root stack over the maximal (or minimal, depending on context) resolution $C^2/G$6 of $C^2/G$7, where $C^2/G$8 is the discriminant divisor.
• Application of the derived McKay correspondence, inducing equivalences $C^2/G$9 compatible with Bridgeland stability, ensuring good behavior of moduli spaces of stable objects.

Bridgeland’s stability conditions on triangulated categories provide a refinement and generalization of classical geometric and representation-theoretic stability notions. Toda’s analysis of the birational geometry (MMP) of surfaces via wall-crossing for Bridgeland moduli is adapted and generalized to the orbifold/root stack context, including explicit constructions of the stability manifold and relevant hearts of t-structures.

Main Results and Technical Contributions

Geometrization of Bridgeland Stability

A principal technical result (Theorem A) asserts that for any smooth contraction $C^3/G$0 of the maximal resolution $C^3/G$1, there exists a connected open subset $C^3/G$2 of the (normalized) stability condition space such that:

• For any $C^3/G$3, the Bridgeland moduli $C^3/G$4 is isomorphic to $C^3/G$5.
• For elementary birational transformations (i.e., single blowup relationships $C^3/G$6), $C^3/G$7 is nonempty and real codimension one.

This provides a direct correspondence between wall-and-chamber decompositions in the stability manifold and the birational models over $C^3/G$8, encoding the entire birational geometry as wall-crossing phenomena of Bridgeland moduli.

Gluing of Geometric and Algebraic Moduli via Derived Equivalence

After identifying the stability manifolds through the derived equivalence, an explicit parameter change, and a rotation action, one finds that the geometric and algebraic local sections of the stability space coincide over their common domain (Theorem B). Thus, the intricate relation between King’s GIT stability parameters and Bridgeland stability on the derived category is made precise in this framework.

This leads directly to a GIT chamber correspondence: for each contraction $C^3/G$9, there is a chamber $G \subseteq \mathrm{SL}_3(\mathbb{C})$0 in the King parameter space $G \subseteq \mathrm{SL}_3(\mathbb{C})$1 such that for generic $G \subseteq \mathrm{SL}_3(\mathbb{C})$2, the moduli space $G \subseteq \mathrm{SL}_3(\mathbb{C})$3 is $G \subseteq \mathrm{SL}_3(\mathbb{C})$4, and adjacent chambers correspond to birational models related by blowups.

Moduli Description and Wall Structure

A notable technical feature is the explicit description of the Bridgeland moduli spaces: for any contraction $G \subseteq \mathrm{SL}_3(\mathbb{C})$5, the moduli consists precisely of objects of the form $G \subseteq \mathrm{SL}_3(\mathbb{C})$6 for $G \subseteq \mathrm{SL}_3(\mathbb{C})$7. The challenge is to recognize these as Bridgeland-stable objects and relate them to King-moduli spaces. The analysis utilizes the interplay of the stability condition, the root stack structure, and the derived McKay equivalence.

The wall-and-chamber structure on the stability manifold is demonstrated to recover the GIT wall-and-chamber decomposition of King stability parameters (Corollary 4.14), and the geometry of these chambers is described in detail for dihedral reflection groups. The results illuminate the structure of the moduli and the behavior under wall-crossing, tying together previously disparate constructions.

Implications and Future Directions

This work demonstrates that Bridgeland stability conditions offer a powerful toolkit for the study of minimal and maximal resolutions in surface quotient singularities, providing a categorically robust alternative to traditional GIT analysis. The results give strong evidence for the general applicability of derived categories and stability manifolds to moduli problems in birational geometry.

The techniques are expected to generalize to other classes of finite groups and quotient singularities, supporting a unified description of the wall-crossing behavior underlying the minimal model program in terms of derived category stability. The derived approach may facilitate the extension to higher-dimensional analogues and to the study of mirror symmetry phenomena, where stability conditions play a key role.

Moreover, the explicit compatibility of stability manifolds under derived equivalence and GIT wall-crossing suggests a deep structural connection between the representation theory of finite groups, noncommutative moduli, and birational algebraic geometry, potentially impacting the formulation and resolution of further open conjectures in the area.

Conclusion

The paper rigorously establishes a new proof of Ishii’s conjecture for dihedral reflection groups, grounded in Bridgeland stability and derived equivalence, offering a conceptual and technical advance over previous approaches. The explicit algebraic and geometric correspondence of moduli spaces, the treatment of wall-crossing and chamber structure, and the extension of previous methods to the orbifold and non-crepant context provide a framework of substantial theoretical value, promising further developments at the interface of algebraic geometry, representation theory, and categorical stability.