[Paper Review] Flops of G-Hilb and equivalences of derived categories by variation of GIT quotient

Alastair Craw, Akira Ishii|arXiv (Cornell University)|Nov 22, 2002

Advanced Algebra and Geometry4 citations

TL;DR

This paper establishes that for finite Abelian subgroups $G \subset \mathrm{SL}(3,\mathbb{C})$, every projective crepant resolution of the quotient singularity $\mathbb{C}^3/G$ arises as a moduli space $\mathcal{M}_\theta$ of $\theta$-stable $G$-constellations via variation of GIT quotient. The key result is a complete classification of such resolutions through the chamber structure of the parameter space $\Theta$, with explicit derived equivalences between moduli spaces in adjacent chambers.

ABSTRACT

For a finite subgroup G in SL(3,C), Bridgeland, King and Reid proved that the moduli space of G-clusters is a crepant resolution of the quotient C^3/G. This paper considers the moduli spaces M_θ, introduced by Kronheimer and further studied by Sardo Infirri, which coincide with G-Hilb for a particular choice of the GIT parameter θ. For G Abelian, we prove that every projective crepant resolution of C^3/G is isomorphic to M_θfor some parameter θ. The key step is the description of GIT chambers in terms of the K-theory of the moduli space via the appropriate Fourier--Mukai transform. We also uncover explicit equivalences between the derived categories of moduli M_θfor parameters lying in adjacent GIT chambers.

Motivation & Objective

• To determine whether every projective crepant resolution of $\mathbb{C}^3/G$ for finite Abelian $G \subset \mathrm{SL}(3,\mathbb{C})$ can be realized as a moduli space $\mathcal{M}_\theta$ via variation of GIT quotient.
• To understand the chamber structure of the parameter space $\Theta$ in terms of $K$-theory and Fourier-Mukai transforms.
• To construct explicit derived equivalences between derived categories of $\mathcal{M}_\theta$ for parameters in adjacent GIT chambers.
• To establish a correspondence between the geometry of crepant resolutions and the stability parameters $\theta$ in the GIT construction.

Proposed method

• Construct moduli spaces $\mathcal{M}_\theta$ and $\overline{\mathcal{M}_\theta}$ of $\theta$-stable and $\theta$-semistable $G$-constellations using geometric invariant theory (GIT).
• Use the derived equivalence $\Phi_C: D(\mathcal{M}_C) \to D^G(\mathbb{C}^3)$ from Bridgeland, King, and Reid to relate $\mathcal{M}_C$ to the equivariant derived category.
• Define the map $L_C: \Theta \to \mathrm{Pic}(\mathcal{M}_C)_{\mathbb{Q}}$ that sends a stability parameter $\theta$ to the ample line bundle $\mathcal{O}_{\mathcal{M}_\theta}(1)$, linking $\theta$ to the ample cone.
• Analyze the chamber decomposition of $\Theta$ via the image of the dual map $\varphi_C^*: \mathrm{Hom}_{\mathbb{Z}}(R(G), \mathbb{Q}) \to K(\mathcal{M}_C)_{\mathbb{Q}}$, identifying $F^1/F^2 \cong \mathrm{Pic}(\mathcal{M}_C)_{\mathbb{Q}}$.
• Use the McKay quiver and representation theory to compute $\mathrm{G\text{-}Ext}^1(Q,S)$, showing its dimension equals the number of connected components of $\mathrm{Band}(Q,S)$, which governs rigidity and deformation.
• Establish that the number of connected components of $\mathrm{Band}(Q,S)$ is at most two, and that connectedness implies rigidity of the quotient sheaf $Q$.

Experimental results

Research questions

• RQ1 Can every projective crepant resolution of $\mathbb{C}^3/G$ for finite Abelian $G \subset \mathrm{SL}(3,\mathbb{C})$ be realized as a moduli space $\mathcal{M}_\theta$ for some GIT parameter $\theta$?
• RQ2 How does the chamber structure of the parameter space $\Theta$ relate to the geometry of the moduli spaces $\mathcal{M}_\theta$?
• RQ3 What is the relationship between the derived categories of $\mathcal{M}_\theta$ for parameters in adjacent GIT chambers?
• RQ4 How do the topological properties of the band $\mathrm{Band}(Q,S)$ determine the rigidity and deformation space of $G$-constellations?
• RQ5 What is the role of the Fourier-Mukai transform in relating $K$-theory of $\mathcal{M}_\theta$ to the representation ring $R(G)$?

Key findings

• Every projective crepant resolution of $\mathbb{C}^3/G$ for finite Abelian $G \subset \mathrm{SL}(3,\mathbb{C})$ is isomorphic to $\mathcal{M}_\theta$ for some GIT parameter $\theta$, proving that all such resolutions arise as moduli spaces of $\theta$-stable $G$-constellations.
• The map $L_C: \Theta \to \mathrm{Pic}(\mathcal{M}_C)_{\mathbb{Q}}$ sends each chamber $C$ to the ample cone of $\mathcal{M}_C$, establishing a direct link between stability parameters and line bundles.
• For parameters in adjacent GIT chambers, the derived categories $D(\mathcal{M}_C)$ and $D(\mathcal{M}_{C'})$ are equivalent via a Fourier-Mukai transform, explicitly constructed from the geometry of the wall crossing.
• The dimension of $\mathrm{G\text{-}Ext}^1(Q,S)$ equals the number of connected components of $\mathrm{Band}(Q,S)$, and this number is at most two, with connectedness implying rigidity of the quotient sheaf $Q$.
• When $\mathrm{Fill}(S)$ is simply connected, the $G$-sheaf $S$ is rigid, as all deformations are trivialized by automorphisms, showing that topological properties of the band control deformation theory.
• The construction provides a complete classification of crepant resolutions of $\mathbb{C}^3/G$ via the chamber decomposition of $\Theta$, with each chamber corresponding to a distinct resolution.