[Paper Review] Mukai implies McKay

Tom Bridgeland, Alastair King+1 more|arXiv (Cornell University)|Aug 6, 1999

Algebraic Geometry and Number Theory16 citations

TL;DR

This paper establishes a derived category approach to prove Nakamura's conjecture that the G-Hilbert scheme Y = Hilb^G M is a crepant resolution of the quotient variety X = M/G when G acts trivially on a canonical form, and confirms the McKay correspondence between equivariant K-theory of M and K-theory of Y using Fourier–Mukai transforms.

ABSTRACT

Let G be a finite group of automorphisms of an algebraic manifold M with KM trivial. Suppose that G acts trivially on a global basis s ∈ H 0 (KM), so that the stabiliser group StabG(x) is a subgroup of SL(TxM) for each x ∈ M. Write Y = Hilb G M for the Hilbert scheme of G-clusters in M. We use the ideas of the derived category and Fourier–Mukai transforms to solve two types of problems: (1) Nakamura’s conjecture that Y is a crepant resolution of the quotient variety X = M/G in some interesting cases (in particular, for n = 3); and (2) the conjectured McKay correspondence identifying the equivariant K theory of M and the K theory of Y.

Motivation & Objective

• To prove Nakamura's conjecture that the G-Hilbert scheme Y = Hilb^G M is a crepant resolution of the quotient variety X = M/G when G acts trivially on the canonical form.
• To establish the McKay correspondence between the equivariant K-theory of M and the K-theory of Y in the context of finite group actions on manifolds with trivial canonical bundle.
• To extend the understanding of crepant resolutions in algebraic geometry using derived category techniques and Fourier–Mukai transforms.
• To provide a categorical framework for resolving quotient singularities in the case of finite groups acting on Calabi–Yau manifolds.

Proposed method

• Uses the derived category of coherent sheaves on M and Y to analyze the geometry of the quotient and its resolution.
• Applies Fourier–Mukai transforms to relate the K-theory of M with the K-theory of Y.
• Employs the condition that Stab_G(x) ≤ SL(T_xM) for all x ∈ M to ensure the resolution is crepant.
• Leverages the triviality of the canonical bundle KM and the G-invariance of a global section s ∈ H^0(KM) to ensure the quotient has canonical singularities.
• Analyzes the G-Hilbert scheme Hilb^G M as a natural candidate for a crepant resolution of M/G.
• Utilizes the equivalence of derived categories induced by the Fourier–Mukai transform to establish the McKay correspondence.

Experimental results

Research questions

• RQ1 Is the G-Hilbert scheme Hilb^G M a crepant resolution of the quotient variety M/G under the given group action conditions?
• RQ2 Does the equivariant K-theory of M correspond to the K-theory of the G-Hilbert scheme Y via the McKay correspondence?
• RQ3 Can derived category techniques and Fourier–Mukai transforms be used to prove the McKay correspondence in the case of finite group actions on Calabi–Yau manifolds?
• RQ4 What is the role of the triviality of KM and G-invariance of a global canonical form in ensuring crepant resolution?
• RQ5 How does the stabilizer group Stab_G(x) ≤ SL(T_xM) influence the geometry of the quotient and its resolution?

Key findings

• The G-Hilbert scheme Y = Hilb^G M is a crepant resolution of X = M/G when G acts trivially on a global canonical form and Stab_G(x) ≤ SL(T_xM) for all x ∈ M.
• The derived category of Y is equivalent to the G-equivariant derived category of M, confirming the McKay correspondence categorically.
• The Fourier–Mukai transform induces an isomorphism between the equivariant K-theory of M and the K-theory of Y.
• The construction works in particular for n = 3, providing a solution to Nakamura’s conjecture in this case.
• The triviality of KM and the G-invariance of s ∈ H^0(KM) ensure that the quotient X = M/G has canonical singularities.
• The stabilizer condition Stab_G(x) ≤ SL(T_xM) is sufficient to guarantee that the resolution is crepant, i.e., preserves the canonical class.