Index theorems on anti-self-dual orbifolds Jeff Viaclovsky August - - PowerPoint PPT Presentation

Introduction Subgroups of SU(2) Cyclic quotient singularities Weighted Projective Spaces

Index theorems on anti-self-dual orbifolds

Jeff Viaclovsky August 6, 2012, Kyoto

Introduction Subgroups of SU(2) Cyclic quotient singularities Weighted Projective Spaces

ALE = Asymptotically Locally Euclidean

Definition

A space (X4, g) is ALE of order τ if there exists a finite subgroup Γ ⊂ SO(4) acting freely on R4 \ {0}, and a diffeomorphism φ : X \ K → (R4 \ B(0, R))/Γ where K is a compact subset of X, satisfying (φ∗g)ij = δij + O(r−τ), and ∂|k|(φ∗g)ij = O(r−τ−k), as r → ∞.

Introduction Subgroups of SU(2) Cyclic quotient singularities Weighted Projective Spaces

Orbifolds

Definition

A Riemannian orbifold (M4, g) is a topological space which is a smooth manifold of dimension 4 with a smooth Riemannian metric away from finitely many singular points. At a singular point p, M is locally diffeomorphic to a cone C on S3/Γ, where Γ ⊂ SO(4) is a finite subgroup acting freely on S3. Furthermore, at such a singular point, the metric is locally the quotient of a smooth Γ-invariant metric on B4 under the orbifold group Γ.

Introduction Subgroups of SU(2) Cyclic quotient singularities Weighted Projective Spaces

Conformal blow-up

Given a compact Riemannian orbifold ( ˆ M, ˆ g) with non-negative scalar curvature, one can use the Green’s function for the conformal Laplacian Gp to associate with any point p a non-compact scalar-flat orbifold by (M \ {p}, gp = G2

pˆ

g). A coordinate system at infinity arises from using inverted normal coordinates in the metric g in a neighborhood of the point p,

Introduction Subgroups of SU(2) Cyclic quotient singularities Weighted Projective Spaces

Conformal blow-down

ALE space (X, g) choose a conformal factor u : X → R+ such that u = O(ρ−2) as ρ → ∞. The space (X, u2g) then compactifies to a C1,α orbifold. In the anti-self-dual case, there moreover exists a C∞-orbifold conformal compactification ( ˆ X, ˆ g) with positive Yamabe invariant. Similarly, a scalat-flat ASD ALE space is ALE of order 2. (Tian-Viaclovsky, Streets, Ache-Viaclovsky).

Introduction Subgroups of SU(2) Cyclic quotient singularities Weighted Projective Spaces

Self-dual or Anti-self-dual metrics

(M4, g) oriented. R =         W + + R

12I

E E W − + R

12I

        . W + = 0 is Anti-self-dual (ASD). W − = 0 is Self-dual (SD). Conformally invariant.

Introduction Subgroups of SU(2) Cyclic quotient singularities Weighted Projective Spaces

ASD Deformation Complex

Γ(T ∗M)

Kg

− → Γ(S2

0(T ∗M)) D

− → Γ(S2

0(Λ2 +)),

where Kg is the conformal Killing operator defined by (Kg(ω))ij = ∇iωj + ∇jωi − 1 2(δω)g, with δω = ∇iωi, S2

0(T ∗M) denotes traceless symmetric tensors,

and D = (W+)′

g is the linearized self-dual Weyl curvature operator.

Introduction Subgroups of SU(2) Cyclic quotient singularities Weighted Projective Spaces

Kuranishi map

The cohomology groups of the above complex yield information about the local structure of the moduli space of anti-self-dual conformal classes. There is a map Ψ : H1(M, g) → H2(M, g) called the Kuranishi map which is equivariant with respect to the action of H0, and the moduli space of anti-self-dual conformal structures near g is locally isomorphic to Ψ−1(0)/H0. Therefore, if H2 = 0, the moduli space is locally isomorphic to H1/H0.

Introduction Subgroups of SU(2) Cyclic quotient singularities Weighted Projective Spaces

Index

The analytical index is given by Ind(M, g) = dim(H0(M, g)) − dim(H1(M, g)) + dim(H2(M, g)), The index is given in terms of topology via the Atiyah-Singer index theorem Ind(M, g) = 1 2(15χ(M) + 29τ(M)), where χ(M) is the Euler characteristic and τ(M) is the signature

• f M.

On an orbifold, there are correction terms to this formula, depending upon the group action.

Introduction Subgroups of SU(2) Cyclic quotient singularities Weighted Projective Spaces

Subgroups of SU(2)

• Type An, n ≥ 1: Γ the cyclic group Zn+1,

exp2πip/(n+1) exp−2πip/(n+1)

• ,

0 ≤ p ≤ n. (1)

• Type Dn, n ≥ 3: Γ the binary dihedral group D∗

n−2 of order

4(n − 2). This is generated by eπi/(n−2) and ˆ j, both acting on the left.

Introduction Subgroups of SU(2) Cyclic quotient singularities Weighted Projective Spaces

Subgroups of SU(2)

• Type E6 : Γ = T∗, the binary tetrahedral group of order 24,

double cover of A(4).

• Type E7 : Γ = O∗, the binary octohedral group of order 48,

double cover of S(4).

• Type E8 : Γ = I∗, the binary icosahedral group of order 120,

double cover of A(5).

Introduction Subgroups of SU(2) Cyclic quotient singularities Weighted Projective Spaces

Index for SU(2) subgroups

Theorem

Let ( ˆ M, ˆ g) be a compact anti-self-dual orbifold with a single

• rbifold point p with orbifold group Γ orientation-preserving

conjugate to type An with n ≥ 1, or Dn with n ≥ 3, or En with n = 6, 7, 8. Then Ind( ˆ M, ˆ g) = 1 2(15χ( ˆ M) + 29τ( ˆ M)) − 4n. (2)

Introduction Subgroups of SU(2) Cyclic quotient singularities Weighted Projective Spaces

Hyperk¨ ahler ALE spaces

Proposition

Let (X, g) be a hyperk¨ ahler ALE space of type An for n ≥ 1, type Dn for n ≥ 3, or type En for n = 6, 7, 8. For −4 ≤ ǫ < 0, let H1

ǫ (X, g) denote the space of traceless symmetric 2-tensors

h ∈ S2

0((T ∗X)) satisfying

(W+)′

g(h) = 0, δg(h) = 0,

with h = O(ρǫ) as ρ → ∞, Then H1

ǫ (X, g) = H1 −4(X, g), and

using the isomorphism S2

0(T ∗M) = Λ2 + ⊗ Λ2 −, H1 −4(X, g) has a

basis {ωI ⊗ ω−

j , ωJ ⊗ ω− j , ωK ⊗ ω− j },

where {ω−

j , j = 1, . . . , n = dim(H2(M))} is a basis of the space

• f L2 harmonic 2-forms.

Introduction Subgroups of SU(2) Cyclic quotient singularities Weighted Projective Spaces

Proof of Proposition

Since Ricci-flat, use that B′(h) = ∆2

Lh = D∗Dh,

to get that infinitesimal ASD deformations are the same as infinitesimal Einstein deformations. Identification of the decaying kernel done in Biquard, Biquard-Rollin, also implicit in earlier work of Itoh.

Introduction Subgroups of SU(2) Cyclic quotient singularities Weighted Projective Spaces

Index comparison

Proposition

Let (X, g) be an anti-self-dual ALE metric with group Γ at infinity, and let ( ˆ X, ˆ g) be the orbifold conformal compactification. Then for −2 < δ < 0, we have Ind( ˆ X, ˆ g) = − dim(H1

δ (X, g)) + dim(H2 −2−δ(X, g))

+ dim(H0(R4/Γ)).

Introduction Subgroups of SU(2) Cyclic quotient singularities Weighted Projective Spaces

Idea of proof of Index comparison

From conformal invariance, together with a removable singularity theorem of Ache-Viaclovsky, dim(H2

−2−δ(X, g)) = dim(H2( ˆ

X, ˆ g)). So the identity becomes dim(H1

δ (X, g)) = dim(H1( ˆ

X, ˆ g)) + {dim(H0(R4/Γ)) − dim(H0( ˆ X, ˆ g))}. The term in braces arises from considering solutions of ω = δKω = 0, which do not extend to global conformal Killing fields.

Introduction Subgroups of SU(2) Cyclic quotient singularities Weighted Projective Spaces

Completion of proof

Kawasaki shows that there must be a formula of the form Ind(M, g) = 1 2(15χ(M) + 29τ(M)) + N where the corrrection term N depending only upon the oriented conjugacy class of the group action, and NOT on the space. Let (X, g) be a Kronheimer ALE metric with orbifold compactification ( ˆ X, ˆ g). Using the above index comparison, we can find N for group action

• rientation-reversing conjugate to ADE-type.

Next, consider a football metric S4/Γ. Then dim H0(S4/Γ) = 15 + N + N′ gives the correction term for orientation-preserving conjugate to ADE-type.

Introduction Subgroups of SU(2) Cyclic quotient singularities Weighted Projective Spaces

LeBrun negative mass metrics

Define gLB = dr2 1 + Ar−2 + Br−4 + r2 σ2

1 + σ2 2 + (1 + Ar−2 + Br−4)σ2 3

• ,

where r is a radial coordinate, and {σ1, σ2, σ3} is a left-invariant coframe on S3 = SU(2), and A = n − 2, B = 1 − n. Redefine the radial coordinate to be ˆ r2 = r2 − 1, and attach a CP1 at ˆ r = 0. After taking a quotient by Zn, with action given by the diagonal action (z1, z2) → exp2πip/n(z1, z2), 0 ≤ p ≤ n − 1, the metric then extends smoothly over the added CP1, is ALE at infinity, and is diffeomorphic to O(−n).

Introduction Subgroups of SU(2) Cyclic quotient singularities Weighted Projective Spaces

A corollary

Corollary

Let ( O(−n), ˆ gLB) be a conformally compactified LeBrun metric. Then Ind( O(−n), ˆ gLB) = 12 − 4n. Consequently, for n ≥ 4, the dimension of the moduli space of anti-self-dual orbifold metrics near a LeBrun metric ( O(−n), ˆ gLB) is at least 4n − 12. Recently Nobuhiro Honda showed that dim = 1 for n = 3, and is equal to 4n − 12 for n ≥ 4. Possible automorphism groups: either trivial or S1, or in the case of n = 4 there is a one-parameter family of SU(2)-invariant ASD metrics. For any n ≥ 3, there is a 1-dimensional family of K¨ ahler scalar-flat deformations of the ALE model.

Introduction Subgroups of SU(2) Cyclic quotient singularities Weighted Projective Spaces

General cyclic action

For 1 ≤ q < p relatively prime integers, we denote by Γ(q,p) the cyclic action exp2πik/p exp2πikq/p

• ,

0 ≤ k < p, Called a type (q, p)-action.

Introduction Subgroups of SU(2) Cyclic quotient singularities Weighted Projective Spaces

Hirzebruch-Jung modified Euclidean algorithm

For 1 ≤ q < p relatively prime integers, write p = e1q − a1 q = e2a1 − a2 . . . ak−3 = ek−1ak−2 − 1 ak−2 = ekak−1 = ek, where ei ≥ 2, and 0 ≤ ai < ai−1, i = 1 . . . k. This can also be written as the continued fraction expansion q p = 1 e1 − 1 e2 − · · · 1 ek . We refer to the integer k as the length of the modified Euclidean algorithm.

Introduction Subgroups of SU(2) Cyclic quotient singularities Weighted Projective Spaces

Index Theorem

Theorem (Lock-V)

Let (M, g) be a compact anti-self-dual orbifold with a single

• rbifold point of type (q, p). The index of the anti-self-dual

deformation complex on (M, g) is given by Ind(M, g) =        1 2(15χtop + 29τtop) +

k

• i=1

4ei − 12k − 2 q = p − 1 1 2(15χtop + 29τtop) − 4p + 4 q = p − 1.

Introduction Subgroups of SU(2) Cyclic quotient singularities Weighted Projective Spaces

Idea of Proof

Directly use Kawasaki’s formula for the correction term using equivariant Chern classes, to arrive at IndΓ( ˆ M) = 1 2(15χtop + 29τtop) − 6 + 14 p

p−1

• j=1
• cot(π

p j) cot(π p qj)

• − 2

p

p−1

• j=1
• cot(π

p j) cot(π p qj) cos(2π p j) cos(2π p qj)

• .

The sum s(q, p) = 1 4p

p−1

• j=1
• cot(π

p j) cot(π p qj)

• is a classical Dedekind sum. No closed expression in general!

Introduction Subgroups of SU(2) Cyclic quotient singularities Weighted Projective Spaces

Some number theory

For x ∈ R, ⌊x⌋ denotes the integer part of x, and {x} = x − ⌊x⌋ denotes the fractional part of x. For x ∈ R \ Z, define the sawtooth function ((x)) = {x} − 1

• 2. We show that the correction term is

N(q, p) = −6 + 12 p

p−1

• j=1

cot(π p j) cot(π p qj) − 4 q−1;p p

• − 4

q p

• ,

where q−1;p is the inverse of q modulo p. We then use classical reciprocity for Dedekind sums, s(q, p) + s(p, q) = −1 4 + 1 12 p q + q p + 1 pq

• together with a new reciprocity for the other terms to deduce the

theorem.

Introduction Subgroups of SU(2) Cyclic quotient singularities Weighted Projective Spaces

Calderbank-Singer ALE spaces

For a type (q, p)-action, the space X is the minimal Hirzebruch-Jung resolution of C2/Γ(q,p), with exceptional divisor given by the union of 2-spheres S1 ∪ · · · ∪ Sk, with intersection matrix (Si · Sj) =        −e1 1 · · · 1 −e2 1 · · · 1 −e3 · · · . . . . . . . . . . . . · · · −ek        , where the ei and k are defined above with ei ≥ 2. Using the Joyce ansatz, Calderbank and Singer produced a (k − 1)-dimensional family of toric ALE scalar-flat K¨ ahler metrics

• n X(q,p).

Introduction Subgroups of SU(2) Cyclic quotient singularities Weighted Projective Spaces

A corollary

Corollary

Let (X, g) be a Calderbank-Singer space with a (q, p)-action at

• infinity. If p > 2, there exist non-toric anti-self-dual ALE

deformations.

Introduction Subgroups of SU(2) Cyclic quotient singularities Weighted Projective Spaces

Weighted Projective Spaces

We first recall the definition of weighted projective spaces in real dimension four:

Definition

For relatively prime integers 1 ≤ r ≤ q ≤ p, the weighted projective space CP2

(r,q,p) is S5/S1, where S1 acts by

(z0, z1, z2) → (eirθz0, eiqθz1, eipθz2), for 0 ≤ θ < 2π. Bryant, and later David-Gauduchon showed there is a canonical Bochner-K¨ ahler metric on CP2

(r,q,p) which is a self-dual K¨

ahler metric. The metric ˜ g = R−2g is Einstein wherever R = 0.

Introduction Subgroups of SU(2) Cyclic quotient singularities Weighted Projective Spaces

Einstein metrics

For a weighted projective space CP2

(r,q,p), there are the following 3

cases:

• When p < r + q the canonical Bochner-K¨

ahler metric has R > 0 everywhere, so it is conformal to a Hermitian Einstein metric with positive Einstein constant.

• When p = r + q the canonical Bochner-K¨

ahler metric has R > 0 except at one point, so it is conformal to a complete Hermitian Einstein metric with vanishing Einstein constant

• utside this point.
• When p > r + q the canonical Bochner-K¨

ahler metric has R vanishing along a hypersurface and the complement is composed of two open sets on which the metric is conformal to a Hermitian Einstein metric with negative Einstein constant.

Introduction Subgroups of SU(2) Cyclic quotient singularities Weighted Projective Spaces

A corollary

Theorem

Let g be the canonical Bochner-K¨ ahler metric with reversed

• rientation on CP

2 (r,q,p). Assume that 1 < r < q < p. Then,

• If p ≤ q + r then [g] is isolated as an anti-self-dual conformal

class.

• If p > q + r then there exist non-trival ASD deformations.

Introduction Subgroups of SU(2) Cyclic quotient singularities Weighted Projective Spaces

Idea of proof

Dedekind sum: s(q, p) = 1 4p

p−1

• j=1
• cot(π

p j) cot(π p qj)

• Rademacher’s triple reciprocity:

s(q−1;rp, r) + s(p−1;qr, q) + s(r−1;pq, p) = −1 4 + 1 12 r pq + q pr + p qr

• ,

We proved a triple reciprocity formula for the other correction terms involving the sawtooth function to obtain an explicit formula for the index in terms of some number-theoretic quantities.

Introduction Subgroups of SU(2) Cyclic quotient singularities Weighted Projective Spaces

Einstein metrics

Using similar ideas, I recently proved

Theorem

If p > 1, then the weighted projective space CP2

(r,q,p) does not

admit any K¨ ahler-Einstein metric with respect to any complex

• structure. Furthermore, if

p ≥ (√q + √r)2, then the weighted projective space CP2

(r,q,p) does not admit any

Einstein metric. Proof is Hitchin-Thorpe type inequality using Dedekind triple reciprocity to compute the orbifold correction term.