Scalar-flat K ahler ALE metrics on minimal resolutions Jeff - - PowerPoint PPT Presentation

Introduction Existence Deformations Other applications

Scalar-flat K¨ ahler ALE metrics on minimal resolutions

Jeff Viaclovsky

University of Wisconsin

May 19, 2015 Vanderbilt University

Jeff Viaclovsky Scalar-flat K¨ ahler ALE metrics on minimal resolutions

Introduction Existence Deformations Other applications

ALE metrics

Definition A complete Riemannian manifold (X4, g) is called asymptotically locally Euclidean or ALE of order τ if there exists a finite subgroup Γ ⊂ SO(4) acting freely on S3 and a diffeomorphism ψ : X \ K → (R4 \ B(0, R))/Γ where K is a compact subset of X, and such that under this identification, (ψ∗g)ij = δij + O(ρ−τ), ∂|k|(ψ∗g)ij = O(ρ−τ−k), for any partial derivative of order k, as r → ∞, where ρ is the distance to some fixed basepoint.

Jeff Viaclovsky Scalar-flat K¨ ahler ALE metrics on minimal resolutions

Introduction Existence Deformations Other applications

Minimal resolutions

Definition Let Γ ⊂ U(2) be a finite subgroup which acts freely on S3. Then, a smooth complex surface ˜ X is called a minimal resolution of C2/Γ if there is a mapping π : ˜ X → C2/Γ such that

1 The restriction π : ˜

X \ π−1(0) → C2/Γ \ {0} is a biholomorphism;

2 π−1(0) is a divisor in ˜

X containing no −1 curves.

Jeff Viaclovsky Scalar-flat K¨ ahler ALE metrics on minimal resolutions

Introduction Existence Deformations Other applications

Minimal resolutions

Definition Let Γ ⊂ U(2) be a finite subgroup which acts freely on S3. Then, a smooth complex surface ˜ X is called a minimal resolution of C2/Γ if there is a mapping π : ˜ X → C2/Γ such that

1 The restriction π : ˜

X \ π−1(0) → C2/Γ \ {0} is a biholomorphism;

2 π−1(0) is a divisor in ˜

X containing no −1 curves.

• These resolutions are minimal in the sense that, given any
• ther resolution πY : Y → C2/Γ, there is a proper analytic

map p : Y → ˜ X such that πY = π ◦ p.

Jeff Viaclovsky Scalar-flat K¨ ahler ALE metrics on minimal resolutions

Introduction Existence Deformations Other applications

Minimal resolutions

Definition Let Γ ⊂ U(2) be a finite subgroup which acts freely on S3. Then, a smooth complex surface ˜ X is called a minimal resolution of C2/Γ if there is a mapping π : ˜ X → C2/Γ such that

1 The restriction π : ˜

X \ π−1(0) → C2/Γ \ {0} is a biholomorphism;

2 π−1(0) is a divisor in ˜

X containing no −1 curves.

• These resolutions are minimal in the sense that, given any
• ther resolution πY : Y → C2/Γ, there is a proper analytic

map p : Y → ˜ X such that πY = π ◦ p.

• For each such Γ, up to isomorphism there exists a unique such

minimal resolution, and in 1968, Brieskorn completely described these resolutions complex-analytically.

Jeff Viaclovsky Scalar-flat K¨ ahler ALE metrics on minimal resolutions

Introduction Existence Deformations Other applications

Existence of scalar-flat K¨ ahler ALE metrics

Theorem (Lock-V) Let Γ ⊂ U(2) be a finite subgroup which acts freely on S3. Then the minimal resolution ˜ X of X = C2/Γ admits scalar-flat K¨ ahler ALE metrics.

Jeff Viaclovsky Scalar-flat K¨ ahler ALE metrics on minimal resolutions

Introduction Existence Deformations Other applications

Small deformations of complex structure

Theorem (Lock-V) Let Γ ⊂ U(2) be a finite subgroup finite subgroup which acts freely

• n S3. Then some small deformations of the minimal resolution ˜

X

• f X = C2/Γ admit scalar-flat K¨

ahler ALE metrics.

Jeff Viaclovsky Scalar-flat K¨ ahler ALE metrics on minimal resolutions

Introduction Existence Deformations Other applications

Existence of extremal K¨ ahler metrics

Theorem (Lock-V) Let Γ ⊂ U(2) be a finite subgroup which acts freely on S3. Then there exist extremal K¨ ahler metrics on certain K¨ ahler classes on the rational surfaces which arise as complex analytic compactifications of the minimal resolution ˜ X of X = C2/Γ.

Jeff Viaclovsky Scalar-flat K¨ ahler ALE metrics on minimal resolutions

Introduction Existence Deformations Other applications

A non-existence result for Ricci-flat metrics

Conjecture of Bando-Kasue-Nakajima: the only simply connected Ricci-flat ALE metrics in dimension four are hyperk¨ ahler.

Jeff Viaclovsky Scalar-flat K¨ ahler ALE metrics on minimal resolutions

Introduction Existence Deformations Other applications

A non-existence result for Ricci-flat metrics

Conjecture of Bando-Kasue-Nakajima: the only simply connected Ricci-flat ALE metrics in dimension four are hyperk¨ ahler. Related to this, Theorem (Lock-V) Let Γ ⊂ U(2) be a finite subgroup which acts freely on S3, and let X be diffeomorphic to the minimal resolution of C2/Γ or any iterated blow-up thereof. If g is a Ricci-flat ALE metric on X, then Γ ⊂ SU(2) and g is hyperk¨ ahler.

Jeff Viaclovsky Scalar-flat K¨ ahler ALE metrics on minimal resolutions

Introduction Existence Deformations Other applications

A non-existence result for Ricci-flat metrics

Conjecture of Bando-Kasue-Nakajima: the only simply connected Ricci-flat ALE metrics in dimension four are hyperk¨ ahler. Related to this, Theorem (Lock-V) Let Γ ⊂ U(2) be a finite subgroup which acts freely on S3, and let X be diffeomorphic to the minimal resolution of C2/Γ or any iterated blow-up thereof. If g is a Ricci-flat ALE metric on X, then Γ ⊂ SU(2) and g is hyperk¨ ahler.

• We do not assume that g is K¨

ahler, only assumption is about the diffeomorphism type.

Jeff Viaclovsky Scalar-flat K¨ ahler ALE metrics on minimal resolutions

Introduction Existence Deformations Other applications

Applications to self-dual examples

Using similar ideas, we can obtain some new examples of self-dual metrics on N#CP2 (the connected sum of N copies of the complex projective plane).

Jeff Viaclovsky Scalar-flat K¨ ahler ALE metrics on minimal resolutions

Introduction Existence Deformations Other applications

Applications to self-dual examples

Using similar ideas, we can obtain some new examples of self-dual metrics on N#CP2 (the connected sum of N copies of the complex projective plane). Theorem (Lock-V) There exist sequences of self-dual metrics on N#CP2 limiting to

• rbifolds with singularities which are not cyclic. These metrics

admit a conformally isometric S1-action.

Jeff Viaclovsky Scalar-flat K¨ ahler ALE metrics on minimal resolutions

Introduction Existence Deformations Other applications

Applications to self-dual examples

Using similar ideas, we can obtain some new examples of self-dual metrics on N#CP2 (the connected sum of N copies of the complex projective plane). Theorem (Lock-V) There exist sequences of self-dual metrics on N#CP2 limiting to

• rbifolds with singularities which are not cyclic. These metrics

admit a conformally isometric S1-action.

• To describe the orbifold groups at the singular points, we need

some background on finite subgroups of SO(4) acting freely

• n S3, which I will describe next.

Jeff Viaclovsky Scalar-flat K¨ ahler ALE metrics on minimal resolutions

Introduction Existence Deformations Other applications

SO(4)

• Identification of C2 and H = {x0 + x1ˆ

i + x2ˆ j + x3ˆ k} : (z1, z2) ∈ C2 ← → z1 + z2ˆ j ∈ H

Jeff Viaclovsky Scalar-flat K¨ ahler ALE metrics on minimal resolutions

Introduction Existence Deformations Other applications

SO(4)

• Identification of C2 and H = {x0 + x1ˆ

i + x2ˆ j + x3ˆ k} : (z1, z2) ∈ C2 ← → z1 + z2ˆ j ∈ H

• Double cover of SO(4) :

φ : S3 × S3 → SO(4) φ(q1, q2)(h) = q1 ∗ h ∗ ¯ q2

Jeff Viaclovsky Scalar-flat K¨ ahler ALE metrics on minimal resolutions

Introduction Existence Deformations Other applications

SO(4)

• Identification of C2 and H = {x0 + x1ˆ

i + x2ˆ j + x3ˆ k} : (z1, z2) ∈ C2 ← → z1 + z2ˆ j ∈ H

• Double cover of SO(4) :

φ : S3 × S3 → SO(4) φ(q1, q2)(h) = q1 ∗ h ∗ ¯ q2

• Double cover of U(2) :

φ : S1 × S3 → U(2)

Jeff Viaclovsky Scalar-flat K¨ ahler ALE metrics on minimal resolutions

Introduction Existence Deformations Other applications

U(2) actions (φ : S1 × S3 → U(2))

• L(q, p) ⊂ U(2) : The subgroup generated by

exp(2πi/p) exp(2πiq/p)

• .

Jeff Viaclovsky Scalar-flat K¨ ahler ALE metrics on minimal resolutions

Introduction Existence Deformations Other applications

U(2) actions (φ : S1 × S3 → U(2))

• L(q, p) ⊂ U(2) : The subgroup generated by

exp(2πi/p) exp(2πiq/p)

• .
• S3 ∼

= SU(2) : Let h1 + h2ˆ j ∈ S3, then (z1 + z2ˆ j) ∗ (h1 + h2ˆ j) ← → h1 −¯ h2 h2 ¯ h1 z1 z2

• .

Jeff Viaclovsky Scalar-flat K¨ ahler ALE metrics on minimal resolutions

Introduction Existence Deformations Other applications

U(2) actions (φ : S1 × S3 → U(2))

• L(q, p) ⊂ U(2) : The subgroup generated by

exp(2πi/p) exp(2πiq/p)

• .
• S3 ∼

= SU(2) : Let h1 + h2ˆ j ∈ S3, then (z1 + z2ˆ j) ∗ (h1 + h2ˆ j) ← → h1 −¯ h2 h2 ¯ h1 z1 z2

• .
• Left quaternionic multiplication:

eiθ ∗ (z1 + z2ˆ j) ← → L(1, p).

Jeff Viaclovsky Scalar-flat K¨ ahler ALE metrics on minimal resolutions

Introduction Existence Deformations Other applications

Finite subgroups

• Γ ⊂ SU(2) : L(−1, p), D∗

4n, T ∗, O∗, I∗

Jeff Viaclovsky Scalar-flat K¨ ahler ALE metrics on minimal resolutions

Introduction Existence Deformations Other applications

Finite subgroups

• Γ ⊂ SU(2) : L(−1, p), D∗

4n, T ∗, O∗, I∗

• Γ ⊂ U(2) which act freely on S3 :

Γ ⊂ U(2) Conditions Order

Jeff Viaclovsky Scalar-flat K¨ ahler ALE metrics on minimal resolutions

Introduction Existence Deformations Other applications

Finite subgroups

• Γ ⊂ SU(2) : L(−1, p), D∗

4n, T ∗, O∗, I∗

• Γ ⊂ U(2) which act freely on S3 :

Γ ⊂ U(2) Conditions Order L(−m, n) (m, n) = 1 n

Jeff Viaclovsky Scalar-flat K¨ ahler ALE metrics on minimal resolutions

Introduction Existence Deformations Other applications

Finite subgroups

• Γ ⊂ SU(2) : L(−1, p), D∗

4n, T ∗, O∗, I∗

• Γ ⊂ U(2) which act freely on S3 :

Γ ⊂ U(2) Conditions Order L(−m, n) (m, n) = 1 n φ(L(1, 2m) × D∗

4n)

(m, 2n) = 1 4mn

Jeff Viaclovsky Scalar-flat K¨ ahler ALE metrics on minimal resolutions

Introduction Existence Deformations Other applications

Finite subgroups

• Γ ⊂ SU(2) : L(−1, p), D∗

4n, T ∗, O∗, I∗

• Γ ⊂ U(2) which act freely on S3 :

Γ ⊂ U(2) Conditions Order L(−m, n) (m, n) = 1 n φ(L(1, 2m) × D∗

4n)

(m, 2n) = 1 4mn φ(L(1, 2m) × T ∗) (m, 6) = 1 24m

Jeff Viaclovsky Scalar-flat K¨ ahler ALE metrics on minimal resolutions

Introduction Existence Deformations Other applications

Finite subgroups

• Γ ⊂ SU(2) : L(−1, p), D∗

4n, T ∗, O∗, I∗

• Γ ⊂ U(2) which act freely on S3 :

Γ ⊂ U(2) Conditions Order L(−m, n) (m, n) = 1 n φ(L(1, 2m) × D∗

4n)

(m, 2n) = 1 4mn φ(L(1, 2m) × T ∗) (m, 6) = 1 24m φ(L(1, 2m) × O∗) (m, 6) = 1 48m

Jeff Viaclovsky Scalar-flat K¨ ahler ALE metrics on minimal resolutions

Introduction Existence Deformations Other applications

Finite subgroups

• Γ ⊂ SU(2) : L(−1, p), D∗

4n, T ∗, O∗, I∗

• Γ ⊂ U(2) which act freely on S3 :

Γ ⊂ U(2) Conditions Order L(−m, n) (m, n) = 1 n φ(L(1, 2m) × D∗

4n)

(m, 2n) = 1 4mn φ(L(1, 2m) × T ∗) (m, 6) = 1 24m φ(L(1, 2m) × O∗) (m, 6) = 1 48m φ(L(1, 2m) × I∗) (m, 30) = 1 120m

Jeff Viaclovsky Scalar-flat K¨ ahler ALE metrics on minimal resolutions

Introduction Existence Deformations Other applications

Finite subgroups

• Γ ⊂ SU(2) : L(−1, p), D∗

4n, T ∗, O∗, I∗

• Γ ⊂ U(2) which act freely on S3 :

Γ ⊂ U(2) Conditions Order L(−m, n) (m, n) = 1 n φ(L(1, 2m) × D∗

4n)

(m, 2n) = 1 4mn φ(L(1, 2m) × T ∗) (m, 6) = 1 24m φ(L(1, 2m) × O∗) (m, 6) = 1 48m φ(L(1, 2m) × I∗) (m, 30) = 1 120m Index–2 diagonal ⊂ φ(L(1, 4m) × D∗

4n)

(m, 2) = 2, (m, n) = 1 4mn

Jeff Viaclovsky Scalar-flat K¨ ahler ALE metrics on minimal resolutions

Introduction Existence Deformations Other applications

Finite subgroups

• Γ ⊂ SU(2) : L(−1, p), D∗

4n, T ∗, O∗, I∗

• Γ ⊂ U(2) which act freely on S3 :

Γ ⊂ U(2) Conditions Order L(−m, n) (m, n) = 1 n φ(L(1, 2m) × D∗

4n)

(m, 2n) = 1 4mn φ(L(1, 2m) × T ∗) (m, 6) = 1 24m φ(L(1, 2m) × O∗) (m, 6) = 1 48m φ(L(1, 2m) × I∗) (m, 30) = 1 120m Index–2 diagonal ⊂ φ(L(1, 4m) × D∗

4n)

(m, 2) = 2, (m, n) = 1 4mn Index–3 diagonal ⊂ φ(L(1, 6m) × T ∗) (m, 6) = 3 24m.

Jeff Viaclovsky Scalar-flat K¨ ahler ALE metrics on minimal resolutions

Introduction Existence Deformations Other applications

Finite subgroups

• Γ ⊂ SU(2) : L(−1, p), D∗

4n, T ∗, O∗, I∗

• Γ ⊂ U(2) which act freely on S3 :

Γ ⊂ U(2) Conditions Order L(−m, n) (m, n) = 1 n φ(L(1, 2m) × D∗

4n)

(m, 2n) = 1 4mn φ(L(1, 2m) × T ∗) (m, 6) = 1 24m φ(L(1, 2m) × O∗) (m, 6) = 1 48m φ(L(1, 2m) × I∗) (m, 30) = 1 120m Index–2 diagonal ⊂ φ(L(1, 4m) × D∗

4n)

(m, 2) = 2, (m, n) = 1 4mn Index–3 diagonal ⊂ φ(L(1, 6m) × T ∗) (m, 6) = 3 24m.

The cases m = 1 are the SU(2) cases.

Jeff Viaclovsky Scalar-flat K¨ ahler ALE metrics on minimal resolutions

Introduction Existence Deformations Other applications

General cyclic action

For 1 ≤ q < p relatively prime integers, recall L(q, p) denotes the cyclic action exp(2πik/p) exp(2πikq/p)

• ,

0 ≤ k < p,

Jeff Viaclovsky Scalar-flat K¨ ahler ALE metrics on minimal resolutions

Introduction Existence Deformations Other applications

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.

Jeff Viaclovsky Scalar-flat K¨ ahler ALE metrics on minimal resolutions

Introduction Existence Deformations Other applications

Continued fraction expansion

This can also be written as the continued fraction expansion q p = 1 e1 − 1 e2 − · · · 1 ek . where ei ≥ 2. We refer to the integer k as the length of the modified Euclidean algorithm.

Jeff Viaclovsky Scalar-flat K¨ ahler ALE metrics on minimal resolutions

Introduction Existence Deformations Other applications

Minimal resolution of cyclic quotients

For a type L(q, p) action, let ˜ X be the minimal resolution of C2/L(q, p).

Jeff Viaclovsky Scalar-flat K¨ ahler ALE metrics on minimal resolutions

Introduction Existence Deformations Other applications

Minimal resolution of cyclic quotients

For a type L(q, p) action, let ˜ X be the minimal resolution of C2/L(q, p). Intersection matrix of curves in the minimal resolution:

✉

−e1

✉

−e2

✉

−ek−1

✉

−ek where the ei and k are defined above with ei ≥ 2.

Jeff Viaclovsky Scalar-flat K¨ ahler ALE metrics on minimal resolutions

Introduction Existence Deformations Other applications

Minimal resolution of cyclic quotients

For a type L(q, p) action, let ˜ X be the minimal resolution of C2/L(q, p). Intersection matrix of curves in the minimal resolution:

✉

−e1

✉

−e2

✉

−ek−1

✉

−ek where the ei and k are defined above with ei ≥ 2.

• Using the Joyce ansatz, Calderbank and Singer (′04) produced

a (k − 1)-dimensional family of toric ALE scalar-flat K¨ ahler metrics on ˜ X.

Jeff Viaclovsky Scalar-flat K¨ ahler ALE metrics on minimal resolutions

Introduction Existence Deformations Other applications

Remarks on examples

• For L(1, p), the cyclic diagonal U(2) action, the

Calderbank-Singer metric is the same as the LeBrun negative mass metric (LeBrun (′88)).

Jeff Viaclovsky Scalar-flat K¨ ahler ALE metrics on minimal resolutions

Introduction Existence Deformations Other applications

Remarks on examples

• For L(1, p), the cyclic diagonal U(2) action, the

Calderbank-Singer metric is the same as the LeBrun negative mass metric (LeBrun (′88)).

• From Γ ⊂ SU(2), hyperk¨

ahler metrics produced and classified by Kronheimer (′89).

Jeff Viaclovsky Scalar-flat K¨ ahler ALE metrics on minimal resolutions

Introduction Existence Deformations Other applications

Remarks on examples

• For L(1, p), the cyclic diagonal U(2) action, the

Calderbank-Singer metric is the same as the LeBrun negative mass metric (LeBrun (′88)).

• From Γ ⊂ SU(2), hyperk¨

ahler metrics produced and classified by Kronheimer (′89).

• Prior to Kronheimer, Ak examples known as Gibbons-Hawking

multi-Eguchi-Hanson metrics (Gibbons-Hawking (′78)).

Jeff Viaclovsky Scalar-flat K¨ ahler ALE metrics on minimal resolutions

Introduction Existence Deformations Other applications

Remarks on examples

• For L(1, p), the cyclic diagonal U(2) action, the

Calderbank-Singer metric is the same as the LeBrun negative mass metric (LeBrun (′88)).

• From Γ ⊂ SU(2), hyperk¨

ahler metrics produced and classified by Kronheimer (′89).

• Prior to Kronheimer, Ak examples known as Gibbons-Hawking

multi-Eguchi-Hanson metrics (Gibbons-Hawking (′78)).

• For L(−1, p), cyclic SU(2) action, the Calderbank-Singer

metric is the same as a multi-Eguchi-Hanson metric with all points on a line (therefore toric).

Jeff Viaclovsky Scalar-flat K¨ ahler ALE metrics on minimal resolutions

Introduction Existence Deformations Other applications

Question

What about scalar-flat K¨ ahler ALE metrics on the minimal resolution of C2/Γ where Γ is one of the following?

Γ ⊂ U(2) Conditions φ(L(1, 2m) × D∗

4n)

(m, 2n) = 1, m > 1 φ(L(1, 2m) × T ∗) (m, 6) = 1, m > 1 φ(L(1, 2m) × O∗) (m, 6) = 1, m > 1 φ(L(1, 2m) × I∗) (m, 30) = 1, m > 1 Index–2 diagonal ⊂ φ(L(1, 4m) × D∗

4n)

(m, 2) = 2, (m, n) = 1 Index–3 diagonal ⊂ φ(L(1, 6m) × T ∗) (m, 6) = 3

Jeff Viaclovsky Scalar-flat K¨ ahler ALE metrics on minimal resolutions

Introduction Existence Deformations Other applications

The main theorem

Theorem (Lock-V) Let Γ ⊂ U(2) be a finite subgroup which acts freely on S3. Then, the minimal resolution of C2/Γ admits scalar-flat K¨ ahler ALE

• metrics. Furthermore, these metrics admit a holomorphic isometric

circle action.

Jeff Viaclovsky Scalar-flat K¨ ahler ALE metrics on minimal resolutions

Introduction Existence Deformations Other applications

Construction

• Start with a LeBrun negative mass metric gLB on (OP1(−2m).

Jeff Viaclovsky Scalar-flat K¨ ahler ALE metrics on minimal resolutions

Introduction Existence Deformations Other applications

Construction

• Start with a LeBrun negative mass metric gLB on (OP1(−2m).
• Scalar-flat K¨

ahler ALE with holomorphic isometry group U(2). (Actually U(2)/Zn where Zn acts on the left.)

Jeff Viaclovsky Scalar-flat K¨ ahler ALE metrics on minimal resolutions

Introduction Existence Deformations Other applications

Construction

• Start with a LeBrun negative mass metric gLB on (OP1(−2m).
• Scalar-flat K¨

ahler ALE with holomorphic isometry group U(2). (Actually U(2)/Zn where Zn acts on the left.)

S³/L(1,2m)

Jeff Viaclovsky Scalar-flat K¨ ahler ALE metrics on minimal resolutions

Introduction Existence Deformations Other applications

Take the quotient (OP1(−2m), gLB)/Γ

S³/L(1,2m) S³/Γ

/Γ L(α , β )

i i

All resulting singularities are of cyclic type!

Jeff Viaclovsky Scalar-flat K¨ ahler ALE metrics on minimal resolutions

Introduction Existence Deformations Other applications

The singularities of (OP1(−2m), gLB)/Γ

S³/Γ

L(α , β )

i i

Γ ⊂ U(2) L(α1, β1) L(α2, β2) L(α3, β3) φ(L(1, 2m) × D∗

4n)

L(1, 2) L(1, 2) L(−m, n) φ(L(1, 2m) × T ∗) L(1, 2) L(−m, 3) L(−m, 3) φ(L(1, 2m) × O∗) L(1, 2) L(−m, 3) L(−m, 4) φ(L(1, 2m) × I∗) L(1, 2) L(−m, 3) L(−m, 5) Index–2 diagonal ⊂ φ(L(1, 4m) × D∗

4n)

L(1, 2) L(1, 2) L(−m, n) Index–3 diagonal ⊂ φ(L(1, 6m) × T ∗) L(1, 2) L(1, 3) L(2, 3)

Jeff Viaclovsky Scalar-flat K¨ ahler ALE metrics on minimal resolutions

Introduction Existence Deformations Other applications

The singularities of (OP1(−2m), gLB)/Γ

They are the same as those which determine the minimal resolution of C2/Γ. S³/Γ

L(α , β )

i i Jeff Viaclovsky Scalar-flat K¨ ahler ALE metrics on minimal resolutions

Introduction Existence Deformations Other applications

Resolving the singularities

• Adapt a gluing theorem for Hermitian anti-self-dual metrics,

due to Rollin-Singer (′05), to attach the appropriate Calderbank-Singer spaces.

• A theorem of Boyer (′86) shows that these Hermitian

anti-self-dual metrics are in fact scalar-flat K¨ ahler.

S³/Γ

Jeff Viaclovsky Scalar-flat K¨ ahler ALE metrics on minimal resolutions

Introduction Existence Deformations Other applications

Resolving the singularities

Theorem Let (M, ω) be a scalar-flat K¨ ahler ALE orbifold of complex dimension 2 with finitely many cyclic singularities satisfying H1(M, R) = 0. Then, the minimal resolution M admits scalar-flat K¨ ahler metrics.

S³/Γ

The proof is nearly identical to that of Rollin-Singer, with the addition of a suitable weight at infinity.

Jeff Viaclovsky Scalar-flat K¨ ahler ALE metrics on minimal resolutions

Introduction Existence Deformations Other applications

Remarks

Remark Recently, Apostolov has shown that the Green’s function metric of a Bochner-K¨ ahler metric on a weighted projective space CP2

(p,q,r)

based at any of the fixed points is K¨ ahler (with respect to a reverse

• riented complex structure). This is scalar-flat, so using such an

ALE orbifold as a starting point, an inductive procedure allows us to obtain examples with any cyclic action at infinity.

Jeff Viaclovsky Scalar-flat K¨ ahler ALE metrics on minimal resolutions

Introduction Existence Deformations Other applications

Remarks

Remark Recently, Apostolov has shown that the Green’s function metric of a Bochner-K¨ ahler metric on a weighted projective space CP2

(p,q,r)

based at any of the fixed points is K¨ ahler (with respect to a reverse

• riented complex structure). This is scalar-flat, so using such an

ALE orbifold as a starting point, an inductive procedure allows us to obtain examples with any cyclic action at infinity. Remark The isometry group of LeBrun’s metrics actually has 2 components, the non-identity component consists of anti-holomorphic isometries. These will be useful later in the construction of self-dual examples.

Jeff Viaclovsky Scalar-flat K¨ ahler ALE metrics on minimal resolutions

Introduction Existence Deformations Other applications

Extremal K¨ ahler metrics on complex analytic compactifications

Using the above, we can also see how to complex analytically compactify the Brieskorn resolutions.

Jeff Viaclovsky Scalar-flat K¨ ahler ALE metrics on minimal resolutions

Introduction Existence Deformations Other applications

Extremal K¨ ahler metrics on complex analytic compactifications

Using the above, we can also see how to complex analytically compactify the Brieskorn resolutions.

• Briefly, one can modify the above construction by quotienting
• ne of Calabi’s extremal K¨

ahler metrics on F2n (Hirzebruch surface), to get 6 orbifold points, of type L(αi, βi), L(−αi, βi), i = 1 . . . 3.

Jeff Viaclovsky Scalar-flat K¨ ahler ALE metrics on minimal resolutions

Introduction Existence Deformations Other applications

Extremal K¨ ahler metrics on complex analytic compactifications

Using the above, we can also see how to complex analytically compactify the Brieskorn resolutions.

• Briefly, one can modify the above construction by quotienting
• ne of Calabi’s extremal K¨

ahler metrics on F2n (Hirzebruch surface), to get 6 orbifold points, of type L(αi, βi), L(−αi, βi), i = 1 . . . 3.

• One can then glue on Calderbank-Singer metrics and resolve

to exremal K¨ ahler metrics on certain K¨ ahler classes on some rational surfaces using the work of Arezzo-Lena-Mazzieri and Arezzo-Pacard-Singer.

Jeff Viaclovsky Scalar-flat K¨ ahler ALE metrics on minimal resolutions

Introduction Existence Deformations Other applications

Small deformations of the complex structure

Theorem (Lock-V) Let Γ ⊂ U(2) be a non-cyclic finite subgroup which acts freely on

• S3. Then, some small deformations of the complex structure on

the minimal resolution of C2/Γ admit scalar-flat K¨ ahler ALE metrics.

Jeff Viaclovsky Scalar-flat K¨ ahler ALE metrics on minimal resolutions

Introduction Existence Deformations Other applications

Idea of proof

• Identify the space of infinitesimal scalar-flat K¨

ahler deformations of (OP1(−2m), gLB) as a representation space of U(2) (Honda (′13)).

Jeff Viaclovsky Scalar-flat K¨ ahler ALE metrics on minimal resolutions

Introduction Existence Deformations Other applications

Idea of proof

• Identify the space of infinitesimal scalar-flat K¨

ahler deformations of (OP1(−2m), gLB) as a representation space of U(2) (Honda (′13)).

• Calculation of the dimension of the space of those

deformations which are invariant under the action of Γ (Actually Γ/Zn).

Jeff Viaclovsky Scalar-flat K¨ ahler ALE metrics on minimal resolutions

Introduction Existence Deformations Other applications

Idea of proof

• Identify the space of infinitesimal scalar-flat K¨

ahler deformations of (OP1(−2m), gLB) as a representation space of U(2) (Honda (′13)).

• Calculation of the dimension of the space of those

deformations which are invariant under the action of Γ (Actually Γ/Zn).

• Honda showed that there are scalar-flat K¨

ahler ALE metrics for all small deformations of O(−n) (which correspond to deforming O(−n) as an affine bundle). Begin construction with deformed metric instead of gLB.

Jeff Viaclovsky Scalar-flat K¨ ahler ALE metrics on minimal resolutions

Introduction Existence Deformations Other applications

Small deformations

Theorem (Lock-V) The dimension of the moduli space of scalar-flat K¨ ahler ALE metrics, dΓ, near a metric obtained above satisfies 0 < 2(bΓ − 1) + kΓ − 3 ≤ dΓ ≤ 2 kΓ

• i=1

(ei − 1)

• + kΓ − 3.
• −bΓ : the self-intersection number of the central rational curve
• −ei : the self-intersection number of the ith rational curve
• kΓ : the number of rational curves in the exceptional divisor

Jeff Viaclovsky Scalar-flat K¨ ahler ALE metrics on minimal resolutions

Introduction Existence Deformations Other applications

Small deformations

Conjecture (Lock-V) For any subgroup Γ ⊂ U(2) which acts freely on S3, all small deformations of the minimal resolution of C2/Γ admit scalar-flat K¨ ahler ALE metrics.

Jeff Viaclovsky Scalar-flat K¨ ahler ALE metrics on minimal resolutions

Introduction Existence Deformations Other applications

Small deformations

Conjecture (Lock-V) For any subgroup Γ ⊂ U(2) which acts freely on S3, all small deformations of the minimal resolution of C2/Γ admit scalar-flat K¨ ahler ALE metrics.

• To prove this, one first needs to understand the existence

problem for scalar-flat K¨ ahler metrics on deformations of Calderbank-Singer metrics.

Jeff Viaclovsky Scalar-flat K¨ ahler ALE metrics on minimal resolutions

Introduction Existence Deformations Other applications

Small deformations

Conjecture (Lock-V) For any subgroup Γ ⊂ U(2) which acts freely on S3, all small deformations of the minimal resolution of C2/Γ admit scalar-flat K¨ ahler ALE metrics.

• To prove this, one first needs to understand the existence

problem for scalar-flat K¨ ahler metrics on deformations of Calderbank-Singer metrics.

• Problem: the deformed complex structures do not admit

holomorphic coordinates at infinity. Develop a gluing procedure for ALE scalar-flat K¨ ahler metrics analogous to Biquard-Rollin (′13).

Jeff Viaclovsky Scalar-flat K¨ ahler ALE metrics on minimal resolutions

Introduction Existence Deformations Other applications

Small deformations

Conjecture (Lock-V) For any subgroup Γ ⊂ U(2) which acts freely on S3, all small deformations of the minimal resolution of C2/Γ admit scalar-flat K¨ ahler ALE metrics.

• To prove this, one first needs to understand the existence

problem for scalar-flat K¨ ahler metrics on deformations of Calderbank-Singer metrics.

• Problem: the deformed complex structures do not admit

holomorphic coordinates at infinity. Develop a gluing procedure for ALE scalar-flat K¨ ahler metrics analogous to Biquard-Rollin (′13).

• Hyperk¨

ahler case can be proved directly using anti-self-dual gluing techniques.

Jeff Viaclovsky Scalar-flat K¨ ahler ALE metrics on minimal resolutions

Introduction Existence Deformations Other applications

A non-existence result for Ricci-flat metrics

Theorem (Lock-V) Let Γ ⊂ U(2) be a finite subgroup which acts freely on S3, and let X be diffeomorphic to the minimal resolution of C2/Γ or any iterated blow-up thereof. If g is a Ricci-flat ALE metric on X, then Γ ⊂ SU(2) and g is hyperk¨ ahler.

Jeff Viaclovsky Scalar-flat K¨ ahler ALE metrics on minimal resolutions

Introduction Existence Deformations Other applications

A non-existence result for Ricci-flat metrics

Theorem (Lock-V) Let Γ ⊂ U(2) be a finite subgroup which acts freely on S3, and let X be diffeomorphic to the minimal resolution of C2/Γ or any iterated blow-up thereof. If g is a Ricci-flat ALE metric on X, then Γ ⊂ SU(2) and g is hyperk¨ ahler.

• See Suvaina (′11), Wright (′11) for a classification of ALE

Ricci-flat K¨ ahler surfaces.

Jeff Viaclovsky Scalar-flat K¨ ahler ALE metrics on minimal resolutions

Introduction Existence Deformations Other applications

Idea of proof

• Applying the equivariant signature theorem to the quotients of

the LeBrun metric described above, we can also compute the η-invariant for any of these finite subgroups, explicitly, in terms of cyclic Dedekind sums. (φ(L(1, 2m) × D∗

4n) case

previously done by Wright (′11).)

Jeff Viaclovsky Scalar-flat K¨ ahler ALE metrics on minimal resolutions

Introduction Existence Deformations Other applications

Idea of proof

• Applying the equivariant signature theorem to the quotients of

the LeBrun metric described above, we can also compute the η-invariant for any of these finite subgroups, explicitly, in terms of cyclic Dedekind sums. (φ(L(1, 2m) × D∗

4n) case

previously done by Wright (′11).) Theorem (Nakajima ’90) Let (X, g) be a Ricci-flat ALE manifold with group at infinity Γ, then 2

• χ(X) − 1

|Γ|

• ≥ 3|τ(X) − η(S3/Γ)|,

with equality if and only if W + or W − vanishes identically.

Jeff Viaclovsky Scalar-flat K¨ ahler ALE metrics on minimal resolutions

Introduction Existence Deformations Other applications

Idea of proof

• Applying the equivariant signature theorem to the quotients of

the LeBrun metric described above, we can also compute the η-invariant for any of these finite subgroups, explicitly, in terms of cyclic Dedekind sums. (φ(L(1, 2m) × D∗

4n) case

previously done by Wright (′11).) Theorem (Nakajima ’90) Let (X, g) be a Ricci-flat ALE manifold with group at infinity Γ, then 2

• χ(X) − 1

|Γ|

• ≥ 3|τ(X) − η(S3/Γ)|,

with equality if and only if W + or W − vanishes identically.

• Detailed analysis of Nakajima-Hitchin-Thorpe inequality using
• ur formula for the η-invariant.

Jeff Viaclovsky Scalar-flat K¨ ahler ALE metrics on minimal resolutions

Introduction Existence Deformations Other applications

Application to self-dual metrics

• The quotient of O(−2m) by ˆ

j acting on the left becomes a non-orientable bundle over RP2. The resulting space is non-K¨ ahler, since ˆ j is an anti-holomorphic map.

Jeff Viaclovsky Scalar-flat K¨ ahler ALE metrics on minimal resolutions

Introduction Existence Deformations Other applications

Application to self-dual metrics

• The quotient of O(−2m) by ˆ

j acting on the left becomes a non-orientable bundle over RP2. The resulting space is non-K¨ ahler, since ˆ j is an anti-holomorphic map.

• Taking a quotient of this by a cyclic group acting on the right,

and resolving the singularities using a gluing theorem for anti-self-dual metrics (instead of scalar-flat K¨ ahler gluing), we can obtain examples with group φ(D∗

4n × L(−1, 2m)) at

infinity.

Jeff Viaclovsky Scalar-flat K¨ ahler ALE metrics on minimal resolutions

Introduction Existence Deformations Other applications

Application to self-dual metrics

• The quotient of O(−2m) by ˆ

j acting on the left becomes a non-orientable bundle over RP2. The resulting space is non-K¨ ahler, since ˆ j is an anti-holomorphic map.

• Taking a quotient of this by a cyclic group acting on the right,

and resolving the singularities using a gluing theorem for anti-self-dual metrics (instead of scalar-flat K¨ ahler gluing), we can obtain examples with group φ(D∗

4n × L(−1, 2m)) at

infinity.

• These can be glued together with a space of type

φ(L(1, 2m) × D∗

4n) at infinity since these groups are

• rientation-reversing conjugate, to obtain the self-dual metrics
• n N#CP2.

Jeff Viaclovsky Scalar-flat K¨ ahler ALE metrics on minimal resolutions

Introduction Existence Deformations Other applications

Application to self-dual metrics

• The quotient of O(−2m) by ˆ

j acting on the left becomes a non-orientable bundle over RP2. The resulting space is non-K¨ ahler, since ˆ j is an anti-holomorphic map.

• Taking a quotient of this by a cyclic group acting on the right,

and resolving the singularities using a gluing theorem for anti-self-dual metrics (instead of scalar-flat K¨ ahler gluing), we can obtain examples with group φ(D∗

4n × L(−1, 2m)) at

infinity.

• These can be glued together with a space of type

φ(L(1, 2m) × D∗

4n) at infinity since these groups are

• rientation-reversing conjugate, to obtain the self-dual metrics
• n N#CP2.
• This construction also works for Index 2 subgroup case.

Jeff Viaclovsky Scalar-flat K¨ ahler ALE metrics on minimal resolutions

Introduction Existence Deformations Other applications

Questions

• Do there exist anti-self-dual ALE metrics with groups at

infinity orientation-reversing conjugate to φ(L(1, 2m) × T ∗), φ(L(1, 2m) × O∗), φ(L(1, 2m) × I∗), and Index 3 subgroup cases?

Jeff Viaclovsky Scalar-flat K¨ ahler ALE metrics on minimal resolutions

Introduction Existence Deformations Other applications

Questions

• Do there exist anti-self-dual ALE metrics with groups at

infinity orientation-reversing conjugate to φ(L(1, 2m) × T ∗), φ(L(1, 2m) × O∗), φ(L(1, 2m) × I∗), and Index 3 subgroup cases?

• Classification of scalar-flat K¨

ahler ALE metrics in dimension 2 using some quotient construction analogous to hyperk¨ ahler quotient construction?

Jeff Viaclovsky Scalar-flat K¨ ahler ALE metrics on minimal resolutions

Introduction Existence Deformations Other applications

Questions

• Do there exist anti-self-dual ALE metrics with groups at

infinity orientation-reversing conjugate to φ(L(1, 2m) × T ∗), φ(L(1, 2m) × O∗), φ(L(1, 2m) × I∗), and Index 3 subgroup cases?

• Classification of scalar-flat K¨

ahler ALE metrics in dimension 2 using some quotient construction analogous to hyperk¨ ahler quotient construction?

• Scalar-flat K¨

ahler ALE metrics in dimensions m > 2?

Jeff Viaclovsky Scalar-flat K¨ ahler ALE metrics on minimal resolutions

Introduction Existence Deformations Other applications

The end

Thank you for your attention.

Jeff Viaclovsky Scalar-flat K¨ ahler ALE metrics on minimal resolutions