Pooya Ronagh
Math 615A - Lie Theory
December 6, 2011
McKay correspondence
1. Kleinian surfaces
The action of SL(2, C) on C2 by left multiplication induces similar action of its subgroups on
C2 :
(
a
c
b
) ∶ (x, y) ↦ (ax + by, xc + dy).
d
The induced action on coordinate ring of C2 is g ∶ f (x, y) ↦ f ○ g −1 hence
a
(
c
b
) ∶ f (x, y) ↦ f (dx − by, −cx + ay).
d
We are looking for the ring of invariant polynomials R = C[x, y]G ⊂ C[x, y]. It is obvious from that
R is generated by homogeneous polynomials.
1.1. Finding the ring of invariants. We need to find a set of homogeneous generators so let us
descend to the action of SL(2, C) on P1 by Mobius transformations. If a homogeneous polynomial
is invariant under the action of G, it in particular satisfies
g.V (f ) = V (g.f ),
∀g ∈ G.
So we will first try to find the ring of such relative invariant polynomials, Rrel . These are invariant
up to multiplication by a character of G:
g.f = χ(g)f, for some χ ∶ G Ð
→ C∗ .
(1) If f is a relative invariant V (f ) is a union of G-orbits so f is a product of relative invariants
with sets of zeroes equal to an orbit; i.e. Rrel is generated by generators of the ideals of
the orbits.
(2) If V (f ) is an orbit with trivial stabilizer group, and p, q ∈ C[x, y] such that V (p) and V (q)
do have nontrivial stabilizer groups. So
g.p = χ1 (g)p,
g.q = χ2 (g)q
where characters χ1 and χ2 have orders e1 and e2 . Therefore for some a, b ∈ C we have
ap∣O1 ∣/e1 + bq ∣O2 ∣/e2
which vanishes on the orbit V (f ). Therefore Rrel is generated by polynomials that cut out
orbits with nontrivial stabilizers!
1
2
2. An example
Let G = BD4n be the finite subgroup of SL(2, C) generated by elements
α=(
ε
0
0
0
) , and β = (
i
ε−1
i
)
0
is an element of order 2n.
where ε = exp 2πi
2n
So we need exceptional orbits of P1 . That means they should be fixed by some g ∈ G. There are
three exceptional orbits in BD4n :
O1 = {0, ∞}, O2 = {n-th roots of 1 }, O2 = { n-th roots of -1 }.
These are cut out by
f1 = xy, f2 = xn − y n , f3 = xn + y n
and respectively have associated characters
χ1 ∶ α ↦ 1, β ↦ −1,
χ2 ∶ α ↦ −1, β ↦ −in ,
χ3 ∶ α ↦ −1, β ↦ in .
Let’s for example take n = 2:
R = C[f22 , f32 , f1 f2 f3 ] = C[x2 y 2 , x2n + y 2n , xy(x2n − y 2n )] = C[U, V, W ]/(U 3 + U V 2 + W 2 ).
2.1. Resolution of the singularity. Now we find a (crepant) resolution of singularities, brought
to us by the Minimal Model Program as a sequence of Blow-ups in this case:
Bl(0,0,0) (V(x3 + xy2 + z2 )):
′
2
2
2
On U1 : ϕ−1
1 Y ∩ U1 = V (u1 + u1 u2 + u3 ). The derivative is (1 + u2 , 2u1 u2 , 2u3 ) and is the zero vector
′
only if u1 u2 = 0 and u3 = 0, which gives points on Y ∩ U if u1 = 0 and u2 = ±i. By i we mean
the solution to the equation 1 + x2 = 0 which if we assume exists in the field gives singularities
(0, 0, 0, 1, ±i, 0) ∈ A3 × P2 or in other words on point [1 ∶ ±i ∶ 0] of the exceptional divisor. I am not
going to blow up these singularities on this chart since U2 will also see these singularities together
with a third one. Hence we will defer blowing up on the three singularities using the single equation
of Y ′ ∩ U2 later on.
′
3
2
2
2
3
2
On U3 : ϕ−1
3 (Y ∩U3 ) = V (u1 u3 +u1 u3 u2 +1) and the differential is (3u1 u3 +u3 u2 , 2u1 u2 u3 , u1 +u1 u2 ) =
2
2
2
2
0 giving singularities only if u1 u2 u3 = 0 and u1 ≠ 0 and u1 + u2 = 3u1 + u2 = 0 which has no
solutions.
′
3
2
2
3
On U2 : ϕ−1
2 (Y ∩ U2 ) = V (u1 u2 + u1 u2 + u3 ) with differentials (3u1 u2 + u2 , u1 + u1 , 2u3 ) which is
3
zero if u3 =, u2 = 0 and u1 + u1 = 0, giving u1 = u2 = u3 = 0 and u1 = ±i, u2 = u3 = 0, and in U2
coordinates for the points (0, 0, 0, 0, 1, 0), (0, 0, 0, ±i, 1, 0). Note that the two points [±i ∶ 1 ∶ 0] are
the same as the singularities found on U1 . We proceed to blowing up this affine variety on (0, 0, 0)
and (±i, 0, 0).
Bl(0,0,0) V(u31 u2 + u1 u2 + u23 ):
′′
2 2
On U21 : with affine chart (w1 , w2 , w3 ) ↦ (w1 , w1 w2 , w1 w3 , 1, w2 , w3 ); ϕ−1
21U21 ∩ Y = V (w1 w + w2 +
2
2
w3 ) with partials (2w1 w2 , w1 + 1, 2w3 ). This vector vanishes only if w3 = w2 = 0 and w1 = ±i. But
these singularities project down to (±i, 0, 0, 1, 0, 0) ↦ (±i, 0, 0). Hence they are lifts of the other two
singularities of Y ′ . So there is no singularity on the exceptional divisor and we are going to blow up
3
these singularities later. Note that since the lift is isomorphic to Y ′ away from the origin it suffices
to resolve the singularities at (±i, 0, 0) on Y ′ itself with appropriate affine chart “centered” at these
points. At least to count the number of blow ups required we can neglect these singularities in this
step.
′′
3
2
2
3
On U22 : ϕ−1
22U22 ∩Y = V (w1 w2 +w1 +w3 ) with partials (3w1 w2 +1, w1 , 2w3 ), vanishing nowhere.
′′
3
2
2
2
3 2
3
On U23 : ϕ−1
23U23 ∩ Y = V (w1 w2 w3 + w1 w2 + 1) with partials (3w1 w2 w3 + w2 , w1 w3 + w1 , 2w1 w2 w3 ).
If it vanishes w1 w2 w3 = 0 hence w2 = 0 hence giving no solution on the surface.
Now we blow-up at (i, 0, 0):
Bl(i,0,0) V(u31 u2 + u1 u2 + u23 ):
We first make a transformation u1 ↦ u1 + i as given by the hint so that we can blowup at the origin.
The new equation is u31 u2 + 3iu21 u2 − 2u1 u2 + u23 = 0:
′
On U21
: again with coordinates (w1 , w2 , w3 ) we get V (w12 w3 +3iw1 w2 −2w2 +w32 ). Of course we need
to show that this polynomial is irreducible in k[w1 , w2 , w3 ] but this can be done using lexicographic
grading of this polynomial ring easily. We will not show the work of it here. The partials are
(2w1 w2 + 3iw2 , w12 + 3iw1 − 2, 2w3 )
which vanishes on the exceptional divisor only if w1 = 0 and it does not. Hence giving no new
singularities.
′
On U22
: The irreducible lift is w13 w22 + 3iw12 w2 − 2w1 + w32 = 0 and the partials are (3w12 w22 + 6iw1 w2 −
2, ⋯) which vanishes only if w2 = 0 giving no singularities on the exceptional divisor.
′
On U23
: The irreducible lift is w13 w2 w32 + 3iw12 w2 w3 − 2w1 w2 + 1 = 0 with first partial 3w12 w2 w32 +
6iw1 w2 w3 − 2w2 which does not vanish on w3 = 0 unless w2 = 0 which gives no points on the
surface.
The blow-up at (−i, 0, 0) is likewise and we will omit showing the computations further.1
3. Finite subgroups of SL(2, C)
Let G be a finite subgroup of the group GL(2, C). It acts on P1 by Mobius transformations.
Let G be a finite subgroup of SL(2, C). Then G is conjugate to a finite subgroup of SU(2). This
is the case because we can take the standard hermitian inner product on C2 and average it by G.
1This figure is copied from Reid’s ‘La correspondence de McKay’.
4
The is a short exact sequence of groups
1
/ Z/2
/ SU(2)
/ SO(3)
_
/1
ϕ
H1
p
/ O(3)
where
ϕ ∶ H1 Ð
→ SU(2),
z
a + bi + cj + dk ↦ ( 1
−z
2
´¹¹ ¸¹ ¶ ´¹¹ ¹ ¹ ¹ ¸¹ ¹ ¹ ¹ ¶
z1
z2
)
z1
(c+di)j=z2 j
and
p ∶ H1 Ð
→ O(3),
q ↦ (x ↦ qxq −1).
It is obvious that p(q) is an orthogonal transformation in R3 . However we may identify the pure
quaternions with S 2 ⊂ R3 and write q ∈ H1 as cos θ + sin θq1 for some q1 ∈ S 2 and observe that p(q)
is the rotation at angle θ about the q1 axis. Therefore p is a surjection onto SO(3).
So any finite subgroup G is a lift of a finite subgroup of SO(3):
(1) The cyclic groups of odd order Cn ⊂ SO(3) are lifted either to an isomorphic Cn ≤ SL(2, C)
or to C2 × Cn ≤ SL(2, C).
(2) Any other finite subgroup of SO(3) contains a non-central element of order 2, hence p ∶
GÐ
→ G has a nontrivial kernel. So we get either a
(a) binary group of even order;
(b) binary dihedral group of order 4n;
(c) binary tetrahedral group of order 24;
(d) binary octahedron group of order 48;
(e) binary icosahedron group of order 120.
All what we have done until now works case-by-case to give a crepant resolution of singularities of
surfaces C2 /G as G varies over all finite subgroups of SL(2, C) classified above:
group
Cn
BD4(n−2)
BT24
BO48
BI120
equation in C3
x2 + y 2 + z n+1
x2 + y 2 z + z n+1
x2 + y 3 + z 4
x2 + y 3 + yz 3
x2 + y 3 + z 5
type of singularity
An
Dn
E6
E7
E8
The intersection matrix of the irreducible components of the exceptional divisor seems to be even
more closely related to the Dynkin diagrams listed above since the self intersection numbers of the
irreducible curves are -2.
5
4. Classical McKay correspondence
Recall that the characters of irreducible representations of G form a basis for the vector space
C[Z(G)], of all functions G Ð
→ C that are constant on conjugacy classes of G. This finite-dimensional
vector space has an hermitian inner product
1
⟨ϕ, ψ⟩ =
∑ ϕ(g)ψ(g).
∣G∣ g∈G
Let ρ ∶ G Ð
→ GL(2, C) = GL(V ) be the faithful representation of G obtained by the embedding
G ↪ SL(2, C). Let {Vi } be the set of irreducible representations of G with characters χi .
The McKay graph (or McKay quiver) of G is the directed multi-graph with vertices Vi and mij edges
from Vi to Vj if Vj occurs mij times in the decomposition of Vi ⊗ V into irreducible representations:
⊕m
Vi ⊗ V = ⊕j Vj ij . Consequently
mij = ⟨χi χρ , χj ⟩.
We find the McKay quivers of the finite groups G as follows: we label the vertices of the McKay
quiver by the dimension of the corresponding irreducible representation di = dim Vi .
Step 0. The cyclic case, Cn , is easy:
For abelian groups the irreducible representations are all 1-dimensional. The number of them is
the same as the number of elements of G. The only possibilities are therefore exp 2πik/n ∶ C Ð
→C
which we denote by ρk . We have ρ = ρ−1 + ρ+1 . And that
ρ ⊗ ρk = ρk−1 + ρk+1 .
̃n :1
This gives Dynkin diagram A
For the other cases we use the following properties:
Proposition 1.
(1) mij = mji ;
(2) The quiver is connected;
(3) mii = 0;
(4) mij ∈ {0, 1}.
Proof.
1This figure is copied from Reid’s ‘La correspondence de McKay’.
6
(1) Note that χρ is real valued since each element of SU(2) has real trace.
mij = ⟨χi χρ , χj ⟩ =
1
1
∑ χi (g)χρ (g)χj (g) =
∑ χi (g)χρ (g)χj (g) = ⟨χi , χρ χj ⟩ = mji .
∣G∣ g∈G
∣G∣ g∈G
(2) This follows from the fact that every irreducible representation of G is contained in some
tensor power of a faithful representation of it.
1
(3) We have mii = ⟨χρ χi , χi ⟩ = ∣G∣
∑g∈G χρ (g)∣χi (g)∣2 . Since G is of even order by our classification above, it contains −id ∈ SU(2). Multiplication by this element gives an automorphism
of G such that χρ (g) = −χρ (−g). Then
2mii =
1
2
2
∑ χρ (g)(∣χi (g)∣ − ∣χi (−g)∣ ) = 0.
∣G∣ g∈G
(4) This is Shwartz inequality:
√
1
1 √
mij =
∑ χi (g)χρ (g)χj (g) ≤
∑ ∣χi (g)∣2 ∣χρ (g)∣2 ∑ ∣χj (g)∣2
∣G∣ g∈G
∣G∣ g∈G
g∈G
√
√
2
2
<
⟨χi , χi ⟩⟨χj , χj ⟩ < 2
∑ ∣χi (g)∣2 ∑ ∣χj (g)∣2 =
∣G∣ g∈G
∣G∣
g∈G
Since dim V = 2, by the above we conclude that 2di = ∑j∼i dj where the sum runs over all adjacent
vertices Vj . We exploit this property:
Step 1. If the is a cycle, the diagram is that of Cn .
Let i1 , ⋯, in+1 be the vertices in some cycle. Then 2di` ≥ di`−1 +di`+1 . We sum over these inequalities
and get an equality. Hence these are the only vertices of the graph labeled by the same dimension.
The trivial representation is of dimension 1, hence all are one dimensional therefore we are in the
cyclic case.
Step 2. No vertex of degree ≥ 5.
Otherwise index them as adjacent vertices, j1 , ⋯, j5 , to i. Then 2di ≥ dj1 + ⋯ + dj5 and also 2djk ≥ di
leading to the contradiction
1
5
di ≤ (dj1 + ⋯ + dj5 ) ≤ di .
4
2
̃4.
Step 3. The only case with a vertex of degree 4 is D
In this case the above argument gives equality
1
di ≤ (dj1 + ⋯ + dj5 ) ≤ di .
2
So these are the only vertices in the graph.
Step 3. There is a vertex of degree 3.
Otherwise the quiver is a path and we check that this cannot happen unless in the cyclic case.
7
Step 4. There is a vertex of degree 3 connected to a leaf of label 1.
̃ n or it is the only vertex of degree 3 in
Step 5. If the vertex of degree 3 has label 2, then we get D
̃6 , E
̃7 , E
̃8 .
which case we get one of the E
Note that if there are at least two vertices, i, j, with degree 3, there are at least x, y, z, w, vertices
connected to i and j and not on a path connecting i to j. Then 2dx , 2dy ≥ di and 2dz , 2dw ≥ dj so
that
di + dj ≥ dx + dy + dz + dw ≥ di + dj
so this has to be an equality.
This completes the proof of the observation of McKay:
Theorem 4.1 (McKay, 1980). The McKay graph of a finite subgroup of SU(2) is one of the
following list of Dynkin diagrams:2
Here the vertex of the trivial representation and that of ρ (in cases it is irreducible) are distinguished
and the vertices are labeled by the dimension of the corresponding irreducible representation.
5. Deeper versions of the correspondence
Let us fix a diagram
M
π
Y
ϕ
/ M /G = X.
2This table is copied from http://web.mit.edu/yisun/www/talks/mckay.pdf.
8
Here M is quasiprojective and KM = 0. G is a finite automorphism group of M acting trivially
on a global section sM ∈ Γ(M, KM ). The orbifold X has trivial canonical class KX = 0 (or trivial
dualizing sheaf ωX = OX ). In view of the Mori theory Y is a crepant resolution of X that need not
be unique in any sense! The Minimal Model Program gives almost a unique one in the M = C2 case.
We will be still talking about this latter case in what follows with the remark that it generalizes to
C3 .
5.1. At the level of homology. [McKay, 1980]
The irreducible components Di of the exceptional locus form a basis {[Di ]} for the homology
H2 (Y, Z). By adding the homology class of a point we have a basis for H∗ (Y, Z) which accounts for
adding the trivial representation. So the McKay correspondence may be stated as a bijection
{ irreducible representations of G } ↔ basis of H∗ (Y, Z) .
5.2. At the level of K-theory. [Gonzalez-Sprinberg-Verdier, 1983]
We work on one hand with the K-theory of Y for the category of vector bundles over it:
K(Y ) = Z ⊕ Pic(Y ), via E ↦ (rk E, c1 (E)),
and on the other hand with the G-equivariant K-theory of C2 which is isomorphic to the module
of representations of G, C[G] via
R(G) Ð
→ KG (C2 ), via ρ ↦ OC2 ⊗C ρ.
Let Z = (Y × XC2 )red be the reduced subscheme of the fiber product
Z

/Z
p
/ C2
q
Y
/X
and we map
Λ ∶ R(G) ≅ KG (C2 ) Ð
→ KG (Z) Ð
→ K(Y ), via ρ ↦ q∗G (p∗ OC2 ⊗C ρ).
Explicitly for the irreducible representations Mi ∶= HomC[G] (Vi , C[x, y]). Then we pull this back to
ϕ∗ Mi ⊇ Ti where T is the OY -submodule of torsion. Then we consider the locally free sheaves
Λ(ρi ) = ϕ∗ Mi /Ti .
Theorem 5.1. The morphism Λ is an isomorphism of abelian groups R(G) ≅ K(Y ) such that
(1) If ρi is irreducible then c1 (Λ(ρi )) = [Ci ] ∈ H 2 (Y, Z);
(2) c1 (Λ(ρi )).c1 (Λ(ρj )) = aij (= Ci ∩ Cj + 2δij );
(3) [ϕ−1(0)] = ∑i (dim ρi )c1 (Λ(ρi )) ∈ H 2 (Y, Z).
9
5.3. At the level of derived categories. [Bridgeland-King-Reid, 2001]
In higher dimensional neither existence nor a canonical choice of Y is guaranteed. The following
properties of the work [GSV] suggests that we want Y to satisfy:
existence of a reduced fiber product Z inheriting a G-action from M with a flat G-invariant morphism q ∶ Z Ð
→ Y of finite degree ∣G∣, in other words we want Z ⊆ Y × C2 be a flat family over Y of
G-equivariant subschemes of M of length ∣G∣.
Nakamura’s G-Hilbert scheme, HilbG (M ) is the irreducible component of the answer to the moduli
problem
F (S) ∶= {Z ⊂ S × M, closed G-invariant flat and finite such that H 0 (OZs ) ≅ C[G]∀s ∈ S}.
containing free G-orbits. The analogue of Z is then the restriction of the universal family over this
moduli space to HilbG (M ):

/ Z p / C2
Z
q
Y
/X
Then B-K-R observed that ‘Mukai implies McKay’:
Theorem 5.2. Let M be a smooth quasi-projective variety, and G ⊆ Aut(M ) a finite group of
automorphisms of M such that
(1) M /G is Gorenstein;
(2) dim Y ×M /G Y ≤ dim Y + 1,
Then HilbG (M ) is a crepant resolution of M /G and the Fourier-Mukai transform
b
Rp∗ ○ q ∗ ∶ Db (Y ) Ð
→ DG
(M )
between the bounded derived category of coherent sheaves on Y and the bounded derived category of
G-equivariant coherent sheaves on M is an equivalence of categories.
This implies the McKay correspondence at the level of K-theory when M = C2 and the conditions
are satisfied for quotient singularities C3 /G with G ⊆ SL(3, C).