2Cm ALF Spaces and Collapsing Ricci-Flat Metrics 0.2Cm 0.1Cm on the K3 Surface

ALF spaces and collapsing Ricci-ﬂat metrics on the K3 surface

Lorenzo Foscolo

Stony Brook University

Simons Collaboration on Special Holonomy Workshop, SCGP, Stony Brook, September 8 2016

September 7, 2016 The Kummer construction as the prototypical example of the formation of orbifold singularities in sequences of Einstein 4–manifolds:

The Kummer construction

Gibbons–Pope (1979): construction of approximately Ricci-ﬂat metrics on 4 the K3 surface. Desingularise T /Z2 by gluing in 16 rescaled copies of the Eguchi–Hanson metric. Rigorous constructions by Topiwala, LeBrun–Singer and Donaldson The Kummer construction

The Kummer construction as the prototypical example of the formation of orbifold singularities in sequences of Einstein 4–manifolds:

A sequence (Mi , gi ) of Einstein 4–manifolds with uniform bounds on Euler characteristic, total volume and diameter converges to an Einstein orbifold M∞ with ﬁnitely many singularities. The geometry around points where the curvature concentrates is modelled by ALE spaces (complete Ricci-ﬂat 4–manifolds with maximal volume growth). Page (1981): consider the Kummer construction along a 1–parameter family 4 3 1 of “split” 4-tori T = T × S` with a circle of length ` → 0. A “periodic but nonstationary” gravitational instanton asymptotic to 3 1 (R × S )/Z2 appears as a rescaled limit. In this talk Page’s suggestion is regarded as a simple case of a more general construction, in which the trivial product T3 × S 1 is replaced by a non-trivial circle bundle over a punctured 3–torus and more general gravitational instantons of cubic volume growth appear as rescaled limits.

4 3 1 The Kummer construction on T = T × S`

In the Ricci-flat case if we drop the diameter bound collapsing can occur: inj gi → 0 everywhere (Anderson, 1992) and the collapse occurs with bounded curvature outside a definite number of points (Cheeger–Tian, 2006). Question: What is the structure of the points where the curvature concentrates? Page (1981): consider the Kummer construction along a 1–parameter family 4 3 1 of “split” 4-tori T = T × S` with a circle of length ` → 0. A “periodic but nonstationary” gravitational instanton asymptotic to 3 1 (R × S )/Z2 appears as a rescaled limit. In this talk Page’s suggestion is regarded as a simple case of a more general construction, in which the trivial product T3 × S 1 is replaced by a non-trivial circle bundle over a punctured 3–torus and more general gravitational instantons of cubic volume growth appear as rescaled limits. 4 Hyperkähler ⇒ Ricci-flat: cr ≤ Vol(Br ) ≤ Cr .

a Minerbe (2010): if Vol Br (p) ≤ Cr with a < 4 for all p ∈ M then a ≤ 3. Moreover, if a = 3 then (M, g) is ALF: There exists a compact set K ⊂ M, R > 0 and a ﬁnite group Γ < O(3) acting freely on S 2 such that M \ K is the total space of a circle ﬁbration 3 over (R \ BR )/Γ. The metric g is asymptotic to a Riemannian submersion

2 −τ 3 g = gR /Γ + θ + O(r ), τ > 0.

If Γ = id then we say that M is an ALF space of cyclic type. If Γ = Z2 we say that M is an ALF space of dihedral type.

ALF gravitational instantons

Gravitational instantons: complete hyperkähler4–manifolds with decaying curvature at infinity( kRmkL2 < ∞. Stronger assumption: faster than quadratic curvature decay, |Rm| = O(r −2−), > 0) Moreover, if a = 3 then (M, g) is ALF: There exists a compact set K ⊂ M, R > 0 and a finite group Γ < O(3) acting freely on S 2 such that M \ K is the total space of a circle fibration 3 over (R \ BR )/Γ. The metric g is asymptotic to a Riemannian submersion

2 −τ 3 g = gR /Γ + θ + O(r ), τ > 0.

If Γ = id then we say that M is an ALF space of cyclic type. If Γ = Z2 we say that M is an ALF space of dihedral type.

ALF gravitational instantons

a Minerbe (2010): if Vol Br (p) ≤ Cr with a < 4 for all p ∈ M then a ≤ 3. ALF gravitational instantons

2 −τ 3 g = gR /Γ + θ + O(r ), τ > 0.

If Γ = id then we say that M is an ALF space of cyclic type. If Γ = Z2 we say that M is an ALF space of dihedral type. 3 Example: x1,..., xn ∈ R , ki ∈ Z>0, λ ≥ 0, n X ki h = λ + 2 |x − xi | i=1

2 Near xi : smooth (orbifold) metric on C /Zki .

At inﬁnity: ALE if λ = 0, ALF if λ > 0.

The Gibbons–Hawking ansatz

Gibbons–Hawking (1978): local form of hyperk¨ahlermetrics in dimension 4 with a triholomorphic circle action. 3 U open subset of R

h positive harmonic function on U

P → U a principal U(1)–bundle with a connection θ

(h, θ) satisﬁes the monopole equation ∗dh = dθ ⇐⇒ gh −1 2 3 g = h gR + h θ is a hyperk¨ahlermetric on P The Gibbons–Hawking ansatz

Gibbons–Hawking (1978): local form of hyperk¨ahlermetrics in dimension 4 with a triholomorphic circle action. 3 U open subset of R

h positive harmonic function on U

P → U a principal U(1)–bundle with a connection θ

(h, θ) satisﬁes the monopole equation ∗dh = dθ ⇐⇒ gh −1 2 3 g = h gR + h θ is a hyperk¨ahlermetric on P 3 Example: x1,..., xn ∈ R , ki ∈ Z>0, λ ≥ 0, n X ki h = λ + 2 |x − xi | i=1

2 Near xi : smooth (orbifold) metric on C /Zki .

At inﬁnity: ALE if λ = 0, ALF if λ > 0. Minerbe (2011): All ALF spaces of cyclic type are multi-Taub–NUT metrics.

In particular, an An ALF metric is asymptotic to the Gibbons–Hawking metric obtained from the harmonic function n + 1 h = 1 + . 2 |x|

ALF spaces of cyclic type

In the previous example, the metrics with ki = 1 for all i and λ > 0 are called multi-Taub–NUT metrics.

A multi-Taub–NUT metric with n + 1 “nuts” is also called an An ALF metric.

The A0 ALF metric is the Taub–NUT metric. ALF spaces of cyclic type

In the previous example, the metrics with ki = 1 for all i and λ > 0 are called multi-Taub–NUT metrics.

A multi-Taub–NUT metric with n + 1 “nuts” is also called an An ALF metric.

The A0 ALF metric is the Taub–NUT metric.

Minerbe (2011): All ALF spaces of cyclic type are multi-Taub–NUT metrics.

In particular, an An ALF metric is asymptotic to the Gibbons–Hawking metric obtained from the harmonic function n + 1 h = 1 + . 2 |x| G. Chen – X. Chen (2015) classiﬁed ALF spaces of dihedral type:

The D0 ALF space is the Atiyah–Hitchin manifold (1988).

The D1 ALF metrics are the double cover of the Atiyah–Hitchin metric and its Dancer deformations (1993).

The D2 ALF spaces are Page’s “periodic but nonstationary” gravitational instantons. Constructed by Hitchin (1984) and Biquard–Minerbe (2011).

Dm, m ≥ 3, constructed by Cherkis–Kapustin (1999) and Cherkis–Hitchin (2005), Biquard–Minerbe (2011) and Auvray (2012).

ALF spaces of dihedral type

(M, g) is a Dm ALF space if up to a double cover g is asymptotic to the Gibbons–Hawking metric obtained from the harmonic function 2m − 4 h = 1 + . 2 |x| ALF spaces of dihedral type

(M, g) is a Dm ALF space if up to a double cover g is asymptotic to the Gibbons–Hawking metric obtained from the harmonic function 2m − 4 h = 1 + . 2 |x|

G. Chen – X. Chen (2015) classiﬁed ALF spaces of dihedral type:

The D0 ALF space is the Atiyah–Hitchin manifold (1988).

The D1 ALF metrics are the double cover of the Atiyah–Hitchin metric and its Dancer deformations (1993).

The D2 ALF spaces are Page’s “periodic but nonstationary” gravitational instantons. Constructed by Hitchin (1984) and Biquard–Minerbe (2011).

Dm, m ≥ 3, constructed by Cherkis–Kapustin (1999) and Cherkis–Hitchin (2005), Biquard–Minerbe (2011) and Auvray (2012). Main Result

Main Theorem (F., 2016)

For every collection of 8 ALF spaces of dihedral type M1,..., M8 and n ALF spaces of cyclic type N1,..., Nn satisfying 8 n X X χ(Mj ) + χ(Ni ) = 24 j=1 i=1 there exists a sequence {g}>0 of KählerRicci-flat metrics on the K3 surface such that: 3 As → 0 the metric g collapses to the flat orbifold T /Z2 with bounded curvature outside 8 + n points. The first 8 points are the fixed points of the involution on T3.

An ALF space of dihedral type arises as a rescaled limit of the sequence close to one of the ﬁxed points of the involution on T3.

An ALF space of cyclic type arises as a rescaled limit of the sequence close to one of the other n points. If Mj is a Dmj ALF gravitational instanton and Ni is an Aki −1 ALF gravitational instanton then the Euler characteristic constraint is equivalent to 8 n X X mj + ki = 16. j=1 i=1

Remarks:

Hitchin (1974) showed that the unique non-flat Ricci-flat 4–manifolds with 2χ + 3τ = 0 are the K3 surface with a hyperkählermetric, an Enriques surface (the quotient of a K3 surface by an involution) with a KählerRicci-flat metric and the quotient of an Enriques surface by an anti-holomorphic involution without fixed points. Luft–Servje (1984) showed that the only flat orientable 3–manifolds 3 admitting an involution with finitely many fixed points are G1 = T , 3 3 G2 = T /Z2 and G6 = T /(Z2 × Z2). Working equivariantly with respect to a finite group action the Main Theorem yields the existence of sequences of Ricci-flat metrics on Enriques surfaces and their quotients by an anti-holomorphic involution collapsing to G2/Z2 and G6/Z2 respectively. Remarks:

If Mj is a Dmj ALF gravitational instanton and Ni is an Aki −1 ALF gravitational instanton then the Euler characteristic constraint is equivalent to 8 n X X mj + ki = 16. j=1 i=1 P P ∗ If mj + ki = 16 then there exists a harmonic function h on T with prescribed singularities: k 2m − 4 h = i + O(1) h = j + O(1) 2 dist(±pi , ·) 2 dist(qj , ·)

Furthermore, there exists a principal U(1)–bundle Mgh → T∗ with a connection θ such that dθ = ∗dh

The GH ansatz over a punctured 3–torus

(T, gT) a ﬂat 3–torus with standard involution τ : T → T Fix(τ) = {q1,..., q8} Assign an integer weight mj ∈ Z≥0 to each qj

Choose further distinct 2n points ±p1,..., ±pn Assign an integer weight ki ≥ 1 to each pair ±pi Denote by T∗ the punctured torus The GH ansatz over a punctured 3–torus

(T, gT) a ﬂat 3–torus with standard involution τ : T → T Fix(τ) = {q1,..., q8} Assign an integer weight mj ∈ Z≥0 to each qj

Choose further distinct 2n points ±p1,..., ±pn Assign an integer weight ki ≥ 1 to each pair ±pi Denote by T∗ the punctured torus P P ∗ If mj + ki = 16 then there exists a harmonic function h on T with prescribed singularities: k 2m − 4 h = i + O(1) h = j + O(1) 2 dist(±pi , ·) 2 dist(qj , ·)

Furthermore, there exists a principal U(1)–bundle Mgh → T∗ with a connection θ such that dθ = ∗dh Remark: The case n = 0 and mj = 2 for all j is the case considered by gh 3 1 Page: M = T × S . Key fact: For > 0 suﬃciently small 1 + h > 0 outside of balls of radius ∝ around the points qj with mj = 0, 1. gh gh Moreover, g descends to a quotient M /Z2. gh gh Goal: For suﬃciently small extend (M /Z2, g ) to a complete metric on a 4–manifold M by gluing

an Aki −1 ALF space in a neighbourhood of ±pi

a Dmj ALF space in a neighbourhood of qj

Background S 1–invariant hyperk¨ahlermetrics

For each > 0 obtain an (incomplete) hyperk¨ahlermetric

gh 2 −1 2 g = (1 + h) gT + (1 + h) θ

gh over the open set M where 1 + h > 0. Key fact: For > 0 suﬃciently small 1 + h > 0 outside of balls of radius ∝ around the points qj with mj = 0, 1. gh gh Moreover, g descends to a quotient M /Z2. gh gh Goal: For suﬃciently small extend (M /Z2, g ) to a complete metric on a 4–manifold M by gluing

an Aki −1 ALF space in a neighbourhood of ±pi

a Dmj ALF space in a neighbourhood of qj

Background S 1–invariant hyperk¨ahlermetrics

For each > 0 obtain an (incomplete) hyperk¨ahlermetric

gh 2 −1 2 g = (1 + h) gT + (1 + h) θ

gh over the open set M where 1 + h > 0.

Remark: The case n = 0 and mj = 2 for all j is the case considered by gh 3 1 Page: M = T × S . gh gh Goal: For suﬃciently small extend (M /Z2, g ) to a complete metric on a 4–manifold M by gluing

an Aki −1 ALF space in a neighbourhood of ±pi

a Dmj ALF space in a neighbourhood of qj

Background S 1–invariant hyperk¨ahlermetrics

For each > 0 obtain an (incomplete) hyperk¨ahlermetric

gh 2 −1 2 g = (1 + h) gT + (1 + h) θ

gh over the open set M where 1 + h > 0.

Remark: The case n = 0 and mj = 2 for all j is the case considered by gh 3 1 Page: M = T × S . Key fact: For > 0 suﬃciently small 1 + h > 0 outside of balls of radius ∝ around the points qj with mj = 0, 1. gh gh Moreover, g descends to a quotient M /Z2. Background S 1–invariant hyperk¨ahlermetrics

For each > 0 obtain an (incomplete) hyperk¨ahlermetric

gh 2 −1 2 g = (1 + h) gT + (1 + h) θ

gh over the open set M where 1 + h > 0.

an Aki −1 ALF space in a neighbourhood of ±pi

a Dmj ALF space in a neighbourhood of qj Every deﬁnite triple ω deﬁnes a Riemannian metric gω: the conformal class is given by Λ+ T ∗M = Span(ω , ω , ω ) gω 1 2 3 1 the volume form is µω = (det Q) 3 µ0

A definite triple ω is said closed if dωi = 0 for i = 1, 2, 3; hyperkähler if it is closed and Q is constant. Donaldson’s question: does every compact 4–manifold with a closed definite triple also admit a hyperkählerstructure?

Deﬁnite triples

Donaldson, 2006: 4 Let (M , µ0) be an oriented 4–manifold with volume form µ0.

A definite triple is a triple ω = (ω1, ω2, ω3) of 2–forms on M such that 2 ∗ Span(ω1, ω2, ω3) is a 3–dimensional positive definite subspace of Λ Tx M at every point x ∈ M. Equivalently the matrix Q defined by 1 ωi ∧ ωj = Qij µ0 is positive definite. 2 A definite triple ω is said closed if dωi = 0 for i = 1, 2, 3; hyperkähler if it is closed and Q is constant. Donaldson’s question: does every compact 4–manifold with a closed definite triple also admit a hyperkählerstructure?

Deﬁnite triples

Donaldson, 2006: 4 Let (M , µ0) be an oriented 4–manifold with volume form µ0.

Every definite triple ω defines a Riemannian metric gω: the conformal class is given by Λ+ T ∗M = Span(ω , ω , ω ) gω 1 2 3 1 the volume form is µω = (det Q) 3 µ0 Donaldson’s question: does every compact 4–manifold with a closed definite triple also admit a hyperkählerstructure?

Deﬁnite triples

Donaldson, 2006: 4 Let (M , µ0) be an oriented 4–manifold with volume form µ0.

Every deﬁnite triple ω deﬁnes a Riemannian metric gω: the conformal class is given by Λ+ T ∗M = Span(ω , ω , ω ) gω 1 2 3 1 the volume form is µω = (det Q) 3 µ0

A definite triple ω is said closed if dωi = 0 for i = 1, 2, 3; hyperkähler if it is closed and Q is constant. Definite triples

Donaldson, 2006: 4 Let (M , µ0) be an oriented 4–manifold with volume form µ0.

Every deﬁnite triple ω deﬁnes a Riemannian metric gω: the conformal class is given by Λ+ T ∗M = Span(ω , ω , ω ) gω 1 2 3 1 the volume form is µω = (det Q) 3 µ0

1 (ω + η ) ∧ (ω + η ) = δ µ . 2 i i j j ij ω Rewrite this as + − − η = F(Q − id + η ∗ η ) and look for η of the form η = da + ζ for ∗ a triple of 1–forms a with d a = 0, and

a triple ζ of harmonic self-dual forms with respect to g . ω

Deforming approximately hyperk¨ahlertriples

gh By gluing ALF spaces to the Gibbons–Hawking hyperkählermanifold M obtain a closed definite triple ω which is approximately hyperkähler: − 1 the matrix Q = (det Q) 3 Q is closer and closer to the identity as → 0.

Question: How to deform ω into a genuine hyperk¨ahlertriple? Deforming approximately hyperk¨ahlertriples

Question: How to deform ω into a genuine hyperk¨ahlertriple? Look for a triple η of closed 2–forms such that

a triple ζ of harmonic self-dual forms with respect to g . ω The linearised equation

We want to apply the Implicit Function Theorem to deform ω to a genuine hyperk¨ahlerstructure. Need to control the norm of the inverse of the linearised operator.

The linearised operator in our problem is

d ∗ + d + :Ω1 → Ω0 ⊕ Ω+.

Consider this operator with respect to the collapsing Gibbons–Hawking gh metric g . As → 0 d ∗ + d + converges to the Dirac operator of the ﬂat 3–torus T3 acting on Z2–invariant spinors (functions and 1–forms).

There is no Z2–invariant kernel! Parameter count

The Main Theorem yields a full-dimensional family of Ricci-ﬂat metrics on the K3 surface:

Flat 3–tori have 6 moduli

3n parameters determine the position of p1,..., pn

The monopole (h, θ) on the punctured torus has 4 moduli (the constant 3 and a point in the dual torus Tˆ parametrising ﬂat connections on T )

The moduli space of Aki −1 ALF metrics has dimension 3(ki − 1)

The moduli space of Dmj ALF metrics has dimension 3mj

n 8 X X 6 + 3n + 4 + 3 (ki − 1) + 3 mj = 58 i=1 j=1 Micallef–Wolfson (2006): there exists a hyperk¨ahlermetric g on the K3 surface and a homology class α such that the area minimiser in α decomposes into the sum of branched minimal immersions, not all of which can be holomorphic with respect to some complex structure compatible with the metric.

Stable minimal surfaces and holomorphicity

Wirtinger’s Inequality: every holomorphic submanifold of a K¨ahler manifold is volume minimising in its homology class.

Micallef (1984): Every stable minimal surface in a ﬂat 4–torus must be holomorphic with respect to some complex structure compatible with the metric.

Question: Can the same result be true for the K3 surface endowed with a hyperk¨ahlermetric? Stable minimal surfaces and holomorphicity

Wirtinger’s Inequality: every holomorphic submanifold of a K¨ahler manifold is volume minimising in its homology class.

Micallef (1984): Every stable minimal surface in a ﬂat 4–torus must be holomorphic with respect to some complex structure compatible with the metric.

Question: Can the same result be true for the K3 surface endowed with a hyperk¨ahlermetric?

Micallef–Wolfson (2006): there exists a hyperk¨ahlermetric g on the K3 surface and a homology class α such that the area minimiser in α decomposes into the sum of branched minimal immersions, not all of which can be holomorphic with respect to some complex structure compatible with the metric. Non-holomorphic strictly stable minimal spheres

An immediate consequence of the Main Theorem is the existence of a (simpler) counterexample:

The D1 ALF space double-cover of the Atiyah–Hitchin manifold contains a strictly stable minimal sphere [Σ] with [Σ] · [Σ] = −4. By the adjunction formula Σ cannot be holomorphic.

Consider the metric g given by the Main Theorem using a conﬁguration of ALF spaces containing the rotationally symmetric D1 ALF space. Use an Implicit Function Theorem for minimal immersions due to White to deform the strictly stable minimal sphere Σ.