[Paper Review] Derived McKay correspondence for GL(3,C)

Yūjirō Kawamata|arXiv (Cornell University)|Sep 29, 2016

Algebraic structures and combinatorial models20 references3 citations

TL;DR

This paper establishes a derived McKay correspondence for finite subgroups of $GL(3,\mathbb{C})$, generalizing Bridgeland-King-Reid's result for $SL(3,\mathbb{C})$. It shows that the equivariant derived category of $\mathbb{C}^3/G$ admits a semi-orthogonal decomposition into derived categories of certain smooth affine varieties $Z_i$ (points, rational curves, or surfaces) and the derived category of a maximal $\mathbb{Q}$-factorial terminalization $Y$ of the quotient singularity $X = \mathbb{C}^3/G$, with the $Z_i$ parametrized by non-trivial inertia subgroups not in $SL(3,\mathbb{C})$. The correspondence extends the DK-hypothesis to non-special linear groups.

ABSTRACT

We prove that the equivariant derived category for a finite subgroup of GL(3,C) has a semi-orthogonal decomposition into the derived category of a certain partial resolution, called a maximal Q-factorial terminalization, of the corresponding quotient singularity and a relative exceptional collection. This is a generalization of a result of Bridgeland, King and Reid.

Motivation & Objective

• To extend the derived McKay correspondence from $SL(3,\mathbb{C})$ to the full $GL(3,\mathbb{C})$ setting.
• To address the failure of the standard equivalence $D^b([\mathbb{C}^3/G]) \cong D^b(Y)$ when $G \not\subset SL(3,\mathbb{C})$ due to non-trivial canonical divisor discrepancies.
• To construct a semi-orthogonal decomposition of $D^b(\text{coh}([\mathbb{C}^3/G]))$ involving derived categories of $Z_i$ and a maximal $\mathbb{Q}$-factorial terminalization $Y$.
• To verify that the components $Z_i$ are rational curves or surfaces, ensuring further semi-orthogonal decompositions into points and curves.

Proposed method

• Define a $\mathbb{Q}$-divisor $B$ on the quotient $X = \mathbb{C}^3/G$ via the pullback condition $\pi^*(K_X + B) = K_{\mathbb{C}^3}$.
• Identify all $G$-invariant proper subspaces $V_j \subset \mathbb{C}^3$ with non-trivial inertia subgroups not contained in $SL(3,\mathbb{C})$, and define decomposition groups $D_j$ and quotients $G_j = D_j/I_j$.
• Construct a maximal $\mathbb{Q}$-factorial terminalization $f: Y \to X$ for the pair $(X, B)$, which is a projective birational morphism with terminal quotient singularities.
• Use the DK-hypothesis to relate inequalities of log canonical divisors to semi-orthogonal decompositions of derived categories along a sequence of toroidal birational maps.
• Apply Theorem 8 to cut out semi-orthogonal components corresponding to irreducible components of $B_Y$ and exceptional loci, yielding derived categories of Deligne-Mumford stacks over $Z_i$.
• Prove that all curves in the decomposition are rational by contradiction using Hochschild homology: if a curve had positive genus, $HH_1(C) \neq 0$, contradicting the derived category structure.

Experimental results

Research questions

• RQ1 Can the derived McKay correspondence be extended from $SL(3,\mathbb{C})$ to $GL(3,\mathbb{C})$ when the group is not special linear?
• RQ2 What replaces the smooth resolution $Y$ in the derived equivalence when $G \not\subset SL(3,\mathbb{C})$?
• RQ3 How do the derived categories of the quotient stack $[\mathbb{C}^3/G]$ decompose when the canonical divisor is not numerically trivial?
• RQ4 What are the geometric components $Z_i$ that appear in the semi-orthogonal decomposition, and what is their dimension and structure?
• RQ5 Are the curves appearing in the decomposition necessarily rational, and how can this be proven categorically?

Key findings

• The equivariant derived category $D^b(\text{coh}([\mathbb{C}^3/G]))$ admits a semi-orthogonal decomposition into $l$ fully faithful images of $D^b(\text{coh}(Z_i))$ and one image of $D^b(\text{coh}(\tilde{Y}))$, where $\tilde{Y}$ is the smooth Deligne-Mumford stack associated to a maximal $\mathbb{Q}$-factorial terminalization $Y$ of $(X, B)$.
• The varieties $Z_i$ are either points (dim 0), smooth rational affine curves (dim 1), or minimal resolutions of quotient surfaces (dim 2), each mapping to $V_j/G_j$ for some $j$.
• When $G \subset SL(3,\mathbb{C})$, the result reduces to the classical Bridgeland-King-Reid equivalence: $D^b([\mathbb{C}^3/G]) \cong D^b(\text{coh}(Y))$, with $Y$ smooth.
• The curves $Z_i$ of dimension 1 are rational, as shown by contradiction: if any had positive genus, Hochschild homology $HH_1(C) \neq 0$, violating the derived category structure.
• The decomposition is not unique, as maximal $\mathbb{Q}$-factorial terminalizations are not unique, but the theorem holds for some choice of such $Y$.
• The components $Z_i$ are globally defined over $X$, even though toroidal structures are étale-local, due to uniqueness of terminalization along generic points of curves.