[Paper Review] Mukai implies McKay: the McKay correspondence as an equivalence of derived categories

Tom Bridgeland, Alastair King+1 more|ArXiv.org|Aug 6, 1999

Advanced Algebra and GeometryMathematics23 references65 citations

TL;DR

This paper establishes the McKay correspondence as an equivalence of derived categories between coherent sheaves on a crepant resolution $ Y $ of a quotient threefold $ X = M/G $ and $ G $-equivariant coherent sheaves on the original threefold $ M $, proving that Nakamura's $ G $-Hilbert scheme is a crepant resolution and that the derived equivalence induces an isomorphism on K-theory. The result generalizes the classical McKay correspondence to higher dimensions using Fourier–Mukai transforms.

ABSTRACT

Let G be a finite group of automorphisms of a nonsingular complex threefold M such that the canonical bundle omega_M is locally trivial as a G-sheaf. We prove that the Hilbert scheme Y=GHilb M parametrising G-clusters in M is a crepant resolution of X=M/G and that there is a derived equivalence (Fourier- Mukai transform) between coherent sheaves on Y and coherent G-sheaves on M. This identifies the K theory of Y with the equivariant K theory of M, and thus generalises the classical McKay correspondence. Some higher dimensional extensions are possible.

Motivation & Objective

• To generalize the classical McKay correspondence beyond dimension 2 by formulating it in terms of derived categories.
• To prove that the $ G $-Hilbert scheme $ \operatorname{G-Hilb}(M) $ is a crepant resolution of $ X = M/G $ when $ M $ is a nonsingular complex threefold and $ G \subset \mathrm{SL}(3,\mathbb{C}) $.
• To establish a derived equivalence (Fourier–Mukai transform) between $ \mathrm{D}(Y) $ and $ \mathrm{D}^G(M) $, where $ Y $ is the $ G $-Hilbert scheme.
• To show that this derived equivalence induces an isomorphism on topological K-theory, explaining the orbifold Euler number conjecture.

Proposed method

• Use the $ G $-Hilbert scheme $ Y = \operatorname{G-Hilb}(M) $ as a candidate for a crepant resolution of $ X = M/G $, focusing on the irreducible component containing free $ G $-orbits.
• Define a Fourier–Mukai transform via the universal $ G $-cluster $ \mathcal{Z} \subset Y \times M $, inducing a functor between derived categories.
• Leverage techniques from Bridgeland’s work on derived categories and stability conditions to prove the equivalence of derived categories.
• Use the fact that the transform is an equivalence if and only if the kernel induces an isomorphism on K-theory and satisfies certain semiorthogonal decomposition conditions.
• Apply topological K-theory and Chern character isomorphisms to relate the derived equivalence to the orbifold Euler number conjecture.
• Verify the correspondence in the Kummer surface case, showing that flat $ G $-line bundles on $ M $ correspond to line bundles supported on $ -2 $-curves on $ Y $.

Experimental results

Research questions

• RQ1 Is the $ G $-Hilbert scheme $ \operatorname{G-Hilb}(M) $ a crepant resolution of $ M/G $ when $ M $ is a nonsingular threefold and $ G \subset \mathrm{SL}(3,\mathbb{C}) $?
• RQ2 Does there exist a derived equivalence between the derived category of coherent sheaves on the resolution $ Y $ and the derived category of $ G $-equivariant coherent sheaves on $ M $?
• RQ3 Can the classical McKay correspondence be generalized to higher dimensions using derived categories and Fourier–Mukai transforms?
• RQ4 Does the derived equivalence induce an isomorphism on topological K-theory, explaining the orbifold Euler number conjecture?
• RQ5 How do flat $ G $-line bundles on $ M $ correspond to line bundles on the resolution $ Y $ in the Kummer surface case?

Key findings

• The $ G $-Hilbert scheme $ \operatorname{G-Hilb}(M) $ is a crepant resolution of $ X = M/G $ for any finite subgroup $ G \subset \mathrm{SL}(3,\mathbb{C}) $, confirming Nakamura’s conjecture.
• There exists a Fourier–Mukai equivalence between $ \mathrm{D}(Y) $ and $ \mathrm{D}^G(M) $, where $ Y $ is the $ G $-Hilbert scheme, establishing the McKay correspondence as a derived equivalence.
• The derived equivalence induces a graded isomorphism between topological K-theory $ \mathcal{K}^*(Y) $ and equivariant K-theory $ \mathcal{K}_G^*(M) $, confirming the orbifold Euler number conjecture $ e(M,G) = e(Y) $.
• In the Kummer surface case, flat $ G $-line bundles on the Abelian surface $ M $ correspond to line bundles $ \mathcal{O}_Y(D(\rho)) $ on the K3 surface $ Y $, where $ D(\rho) = \frac{1}{2} \sum \rho(x_i) C_i $, with $ C_i $ the $ -2 $-curves.
• The derived category approach provides a uniform, case-free proof of the existence of crepant resolutions in dimension 3, bypassing the need for case-by-case analysis.
• The correspondence on K-theory is an isomorphism for all finite subgroups of $ \mathrm{SL}(3,\mathbb{C}) $, generalizing earlier results for abelian groups.