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Surfaces Moduli K3 surfaces Algebraic geometry Details

Collapsing Ricci-flat metrics on K3 surfaces

Jeff Viaclovsky

University of California, Irvine

June 27, 2018 Tokyo Institute of Technology

Jeff Viaclovsky Collapsing Ricci-flat metrics on K3 surfaces

Surfaces Moduli K3 surfaces Algebraic geometry Details

Surfaces

Compact orientable surfaces are classified by their genus γ: γ = 0 γ = 1 γ ≥ 2.

Jeff Viaclovsky Collapsing Ricci-flat metrics on K3 surfaces

Surfaces Moduli K3 surfaces Algebraic geometry Details

Metrics and distance

Definition A Riemannian metric g on a manifold M is a smooth choice of positive definite inner product on each tangent space TpM. This gives us a way to compute distances, since the length of a path α : [a, b] → M is given by L(α) = b

a

• g( ˙

α, ˙ α)dt. The distance between points p1 and p2 in the infimum L over all smooth paths connecting p1 and p2.

Jeff Viaclovsky Collapsing Ricci-flat metrics on K3 surfaces

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Volumes

Let e1 and e2 be an ONB of the tangent space TpM, and let e1 and e2 denote the dual basis of T ∗

p M. That is,

e1(e1) = 1, e1(e2) = 0 e2(e1) = 0, e2(e2) = 1. The form e1 ∧ e2 is a well-defined 2-form, independent of the choice of basis, which is called the volume form, and we let dVg ≡ e1 ∧ e2. The volume of a region U ⊂ M is defined by V ol(R) =

• U

dVg.

Jeff Viaclovsky Collapsing Ricci-flat metrics on K3 surfaces

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Gaussian Curvature

Definition The Gaussian curvature K : M → R is defined by V ol(B(p, r)) = πr2 1 + K(p) 12 r2 + O(r4)

• ,

as r → 0, where B(p, r) is a ball of radius r centered at p.

Jeff Viaclovsky Collapsing Ricci-flat metrics on K3 surfaces

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Constant Gauss curvature

A “best” metric on a surface is one which has constant curvature K =      1 spherical flat −1 hyperbolic. It turns out that such a metric always exists on a compact surface, this is part of the uniformization theorem.

Jeff Viaclovsky Collapsing Ricci-flat metrics on K3 surfaces

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The sphere

The flat metric in R3 restricts to a very nice metric on the unit sphere in R3, called the round metric. The round metric is the unique “best” metric on the 2-sphere: Theorem If (S2, g) is any Riemannian metric on the 2-sphere with constant curvature 1, then there exists a diffeomorphism ϕ : S2 → S2 so that ϕ∗g is the standard round metric.

Jeff Viaclovsky Collapsing Ricci-flat metrics on K3 surfaces

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The torus

The usual picture of a torus (as a surface of revolution in R3) does not represent a “best” metric. Instead, we define it as (T 2, g) =

• R2/Z ⊕ Z, g0
• where the action is by integer translations in either coordinate

direction, and g0 is the flat metric, so K ≡ 0.

Jeff Viaclovsky Collapsing Ricci-flat metrics on K3 surfaces

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Lattices

More generally, we can consider a lattice in R2 generated by {1, τ} where τ is a complex number in the upper half plane. Consider the quotient (T 2, g) =

• R2/Z ⊕ Z · τ, g0
• .

For any τ in the upper half plane, we get a flat metric on the torus.

Jeff Viaclovsky Collapsing Ricci-flat metrics on K3 surfaces

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Moduli of flat metrics

It turns out that the lattices determined by τ and τ ′ determine the same flat metric on T 2 if and only if τ ′ = aτ + b cτ + d, where a b c d

• ∈ SL(2, Z)/{±1}.

Consequently, there is a 2-dimensional family of flat metrics on a torus.

Jeff Viaclovsky Collapsing Ricci-flat metrics on K3 surfaces

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Moduli of flat metrics

The above family contains all flat metrics on a torus (up to diffeomorphism): Theorem If g is any flat metric on the torus T 2, then there exists a diffeomorphism ϕ : T 2 → T 2 such that ϕ∗g is the quotient of the flat metric on R2 by a lattice. So, up to scaling, there is a 2-dimensional moduli space of solutions to the equation K = 0.

Jeff Viaclovsky Collapsing Ricci-flat metrics on K3 surfaces

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Compactification

There are 2 ways to compactify this moduli space.

• Scaling to unit diameter, a torus limits to a circle, with only
• ne of the S1 directions shrinking. (We say that the torus

limits to the circle in the Gromov-Hausdorff sense.

• Algebraically, we can add a singular elliptic curve, a nodal

cubic curve

Figure: y2 = x3 − x2

Jeff Viaclovsky Collapsing Ricci-flat metrics on K3 surfaces

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Higher genus

The Gauss-Bonnet theorem states that

• M

KdVg = 2πχ(M), where χ(M) = V − E + F is the Euler characteristic. If K = −1, then χ(M) = 2 − 2γ < 0, so the genus is ≥ 2. Riemann showed that there is a moduli space of solutions to the equation K = −1 of dimension dimR(Mγ) = 6γ − 6 Compactification is more involved, involving limits with hyperbolic cusps.

Jeff Viaclovsky Collapsing Ricci-flat metrics on K3 surfaces

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Scalar curvature

In higher dimensions, the expansion of volume of a ball is V ol(B(p, r)) = ωnrn 1 − R(p) 6(n + 2)r2 + O(r4)

• ,

as r → 0. This defines a curvature quantity R called the scalar curvature.

Jeff Viaclovsky Collapsing Ricci-flat metrics on K3 surfaces

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Ricci Curvature

The expansion of the volume element in normal coordinates is dVg =

• 1 − 1

6Rklxkxl + O(r3)

• dx1 ∧ · · · ∧ dxn,

as r → 0. This defines a tensor Ric = Rkldxk ⊗ dxl, called the Ricci tensor.

Jeff Viaclovsky Collapsing Ricci-flat metrics on K3 surfaces

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Einstein metrics

The condition that Ric(g) = λ · g, for some constant λ, is a generalization of the constant Gaussian curvature case to higher dimensions, and solutions are known as Einstein metrics. Einstein was interested in solutions of this equation on a Lorentzian manifold (with metric g of signature (1, 3)), in which case the equations are hyperbolic. But in recent years mathematicians have become interested in solutions on a Riemannian manifold (with a positive definite metric), and such solutions are also very important in physics.

Jeff Viaclovsky Collapsing Ricci-flat metrics on K3 surfaces

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Moduli of Einstein metrics

For a manifold M of dimension n, the moduli space of solutions is defined by Mλ = {g ∈ Met(M) | Ric(g) = λ · g}/Diff(M) We must mod out by the infinite dimensional group Diff(M) of diffeomorphisms of M, which makes things tricky. The trick is to show that there is a “slice” for the diffeomorphism group action on the space of metrics (Ebin). Then Einstein’s equations become elliptic. The only failure of ellipticity comes from the diffeomorphism action directions. (Details omitted).

Jeff Viaclovsky Collapsing Ricci-flat metrics on K3 surfaces

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Ellipticity

The equation Ric(g) = λ · g, is nonlinear, and, modulo the diffeomorphism directions, very roughly looks like ∆g = g ∗ ∇g so is “quasilinear elliptic”.

Jeff Viaclovsky Collapsing Ricci-flat metrics on K3 surfaces

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The linearized operator

The moduli space of Einstein metrics near a given Einstein metric can be studied using the linearized operator. Indeed, if we write F(θ) = Ric(g + θ) − λ · (g + θ), then F ′(h) = d dtF(th)

• t=0

After getting rid of the diffeomorphism directions, the linearized

• perator is of the form

F ′(h) = ∆h + lower order terms, where the lower order terms involve the curvature of g, so is an elliptic operator.

Jeff Viaclovsky Collapsing Ricci-flat metrics on K3 surfaces

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Lyapunov-Schmidt reduction

One would like to say that the moduli space of Einstein metrics looks like the kernel of the linearized operator. However, since the

• perator is nonlinear, this may or may not be true. What is true is

the following: Theorem There exists a mapping Ψ : Ker(F ′) → Coker(F ′) such that the moduli space of Einstein metrics near g looks like Ψ−1(0). That is, the zero space of the operator F mapping between between infinite-dimensional spaces looks exactly like the zero set

• f a smooth map between finite-dimensional spaces.

Jeff Viaclovsky Collapsing Ricci-flat metrics on K3 surfaces

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Fixed Point Theorem

The idea of Lyapunov-Schmidt reduction is the following. For each x ∈ Ker(F ′), F(x + h) = 0 has an expansion F(0) + F ′(x + h) + Q(x + h) = 0, where Q(h) are the nonlinear terms. If F ′ is surjective with bounded right inverse G, then writing h = Gy, we have y + Q(x + Gy) = 0.

Jeff Viaclovsky Collapsing Ricci-flat metrics on K3 surfaces

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Fixed Point Theorem II

A solution y of y + Q(x + Gy) = 0, is a fixed point of a mapping T : y → −Q(x + Gy). For x and y sufficiently small, this is a contraction mapping, so for any initial y0, the sequence y0, Ty0, T 2y0, . . . , T ky0, . . . will converge to a unique solution. (The general case is then proved by projecting to the orthogonal complement of the Cokernel.)

Jeff Viaclovsky Collapsing Ricci-flat metrics on K3 surfaces

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Bubbling

In general, a sequence of Einstein metrics can limit to a metric space with singularities, and “bubbling” happens at the points of singularity formation. Convergence is in the Cheeger-Gromov-Hausdorff topology (metric space convergence):

Jeff Viaclovsky Collapsing Ricci-flat metrics on K3 surfaces

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Gluing

Reversing this process is sometimes called “gluing.”. That is, given a solution of your equation with a singularity, and a possible “bubble”, can you somehow “glue” the bubble on, and then perturb your approximate solution to an exact solution of the equation? Theorem There exists a mapping Ψ : space of “gluing parameters” → space of “obstructions” such that the space of solutions near the approximate metric is given by Ψ−1(0). Thus the infinite dimensional question is reduced to a finite-dimensional question, and if the obstruction space vanishes, then the gluing can be carried out.

Jeff Viaclovsky Collapsing Ricci-flat metrics on K3 surfaces

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K¨ ahler metrics

A K¨ ahler manifold is a very special type of complex manifold. Definition A complex manifold is a real manifold which can be covered by coordinate charts with holomorphic overlap maps. A K¨ ahler manifold is a complex manifold with a Riemannian metric ω which can be written locally as ω = √ −1∂ ¯ ∂ϕ where ϕ : M → R is a function. Intuitively, a metric on a complex manifold is K¨ ahler if the Riemannian geometry and complex geometry coincide.

Jeff Viaclovsky Collapsing Ricci-flat metrics on K3 surfaces

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Yau’s Theorem

Theorem (Yau 1976) A compact K¨ ahler manifold admits a Ricci-flat K¨ ahler metric ⇐ ⇒ c1(X) = 0. Abstract existence theorem. What do metrics looks like? Natural families:

• complex structure J
• K¨

ahler class [ω].

Jeff Viaclovsky Collapsing Ricci-flat metrics on K3 surfaces

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K3 surfaces

X = {f4(z0, z1, z2, z3) = 0} ⊂ P3. Algebraic K3s: 19-dimensional family. Since KX is trivial, H1(X, Θ) ≡ H1(X, Ω1) so there is actually a b1,1 = 20-dimensional family of Js. Each J has a 20-dimensional K¨ ahler cone. Moduli of Yau’s metrics= 40 + 20 = 60-dimensional? Overcounted:

Jeff Viaclovsky Collapsing Ricci-flat metrics on K3 surfaces

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Hyperk¨ ahler struture

K¨ ahler = ⇒ Hol ⊂ U(2). KX trivial = ⇒ ∃Ω = ωJ + iωK parallel (2, 0)-form = ⇒ Hol ⊂ Sp(1) = SU(2). Each of Yau’s metrics is K¨ ahler w.r.t, aI + bJ + cK, a2 + b2 + c2 = 1, an S2s worth of complex structures. Metric moduli = 58-dimensional.

Jeff Viaclovsky Collapsing Ricci-flat metrics on K3 surfaces

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General theory

Ric(gj) = 0 = ⇒ Gromov-Hausdorff limit.

• Singularity formation =

⇒ curvature blows up.

• Bubbling phenomena: rescaled limits are complete Ricci-flat

spaces.

• Volume non-collapsing: V ol(Bpj(1)) > v0 > 0 =

⇒ orbifold limit.

• Volume collapsing V ol(Bpj(1)) → 0 =

⇒ lower-dimensional limit. Theorem (Cheeger-Tian) Sequence collapses with uniformly bounded curvature away from finitely many points.

Jeff Viaclovsky Collapsing Ricci-flat metrics on K3 surfaces

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Examples produced by gluing techniques

• Kummer surface: 4-dim limit = T 4/Z2, with flat metric. At

16 singular points, Eguchi-Hanson metric on OP1(−2) bubbles

• ff. Bubbles are ALE.
• Foscolo: 3-dim limit = T 3/Z2, with flat metric. At 8 singular

points, ALF D2 metrics bubble off.

• Gross-Wilson: 2-dim limit = S2. Away from 24 singular

points, sequence collapses with uniformly bounded curvature, with T 2-fibers being uniformly scaled down. At 24 singular points, Taub-NUT ALF metrics bubble off.

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Chen-Chen

Chen-Chen: 1-dim limit = [0, 1]. Singular points at 0 and 1. Interior: collapse with unformly bounded curvature, uniform shrinking of flat T 3. Bubbles are ALH spaces: g = dr2 + gT 3 + O(e−δr). as r → ∞, which arise from rational elliptic surfaces: RES = Blp1,...,p9P2

π

− → P1, and X = RES \ T 2, where T 2 is a smooth fiber (Tian-Yau). Chen-Chen produce these examples by gluing together 2 ALH factors with a long cylindrical region in between, using earlier ideas

• f Kovalev-Singer, Floer.

Jeff Viaclovsky Collapsing Ricci-flat metrics on K3 surfaces

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Tian-Yau metrics

Let DPb be a degree 1 ≤ b ≤ 9 del Pezzo surface. Let T 2 ⊂ DPb be a smooth anticanonical divisor. Theorem (Tian-Yau) Xb = DPb \ T 2 admits a complete Ricci-flat K¨ ahler metric, which is asymptotic to a Calabi ansatz metric on a punctured disc bundle in NT 2. Solution of the form ωg =

i 2π

• ∂∂(− log S2)

3 2 + ∂∂φ

• .

We would like to “glue” two of these spaces together, but the asymptotic geometry is not cylindrical: need to find appropriate neck region.

Jeff Viaclovsky Collapsing Ricci-flat metrics on K3 surfaces

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Main result

Theorem (Hein-Sun-Viaclovsky-Zhang) Given integers 1 ≤ b± ≤ 9 and 1 ≤ m ≤ b+ + b−, there is a family

• f hyperk¨

ahler metrics gβ on a K3 surface which collapse to an interval [0, 1], (K3, gβ) GH − − → ([0, 1], dt2), β → ∞, such that the following topological and regularity properties hold.

• There exist distinct points ti ∈ (0, 1), i = 1 . . . m, such that

at fixed distance away from the ti, the sequence collapses with uniformly bounded curvature, with regular fibers diffeomorphic to 3-dimensional Heisenberg nilmanifolds or 3-dimensional tori.

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Main result cont’d

Theorem (HSVZ cont’d)

• There exist points xβ,i → ti, such that |Rmgβ|(xβ,i) → ∞ as

β → ∞, and rescalings of the metrics near xβ,i converge to Taub-NUT metrics.

• If t = 0 or t = 1, there exist points xβ,i → t, such that

|Rmgβ|(xβ,i) → ∞ as β → ∞, and rescalings of the metrics near xβ,i converge to Tian-Yau metrics on del Pezzo surfaces

• f degree b± minus an anticanonical elliptic curve.

By varying the choice of neck region, we can arrange that the number of singular points in the interior can be any integer in [1, b− + b+]. Also, the degrees of the nilmanifolds in the regular collapsing regions can vary from −b+ to b− and all such degrees can occur.

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Heisenberg nilmanifolds

Nil3

b is an S1-bundle over T 2 of degree b:

S1

Nil3

b π

• T 2.

In our main theorem, in the regular collapsing regions, the T 2s and the S1s shrink at different rates: diam(T 2) ∼ β−1 diam(S1) ∼ β−2.

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The approximate metric

× × × × × × × × × × × × × × Xb− Xb+ N t1 1

Figure: The vertical arrows represent collapsing to a one-dimensional

• interval. The red circles represent the S1 fibers, the blue curves represent

the T 2s, and the ×s are the Taub-NUTs. The gray regions are in the damage zones.

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Models for degenerations of complex structure

• Let b+ = b− = 9. X+ = X− = P2 \ {s3 = 0}
• X = degree 2 K3 surface: π : X → P2, 2 : 1 branched over a

sextic s6.

• Degeneration: s6 → s2

3

• This case was previously conjectured by R. Kobayashi.

Jeff Viaclovsky Collapsing Ricci-flat metrics on K3 surfaces

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Models for degenerations of complex structure

• Let b+ = b− = 8, with DP8 = S2 × S2.

Q+ = {q+ = 0} ⊂ P3 Q− = {q− = 0} ⊂ P3 Q+ ∩ Q− = T 2. X+ = Q+ \ T 2, X− = Q− \ T 2.

• Degeneration: smooth quartic q4 → q2 · q′
• 2. Neck is a

desingularization of the union of 2 nonsingular quadrics.

Jeff Viaclovsky Collapsing Ricci-flat metrics on K3 surfaces

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Models for degenerations of complex structure

• Let b+ = 3, b− = 9, with

DP3 = Blp1,...,p6P2 = {q3 = 0} ⊂ P3. DP9 = P2 = {l1 = 0} ⊂ P3 DP3 ∩ DP9 = T 2. X+ = DP3 \ T 2, X− = P2 \ T 2.

• Degeneration: smooth quartic q4 → q3 · l1. Neck is a

desingularization of the union of a plane and a nonsingular cubic.

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Models for degenerations of complex structure

Conjecture The only possible Gromov-Hausdorff limits of Ricci-flat metrics on K3 surfaces are

• Ricci-flat orbifolds with ADE singularities.
• T 4/Z2 with a flat metric.
• T 3/Z2 with a flat metric.
• S2 (with some possibly singular Hessian metric).
• ([0, 1], dt2).

Related to work of Odaka-Oshima, relating Gromov-Hausdorff limits of Ricci-flat metrics on K3 surfaces to complex degenerations, and boundary components of the compactification

• f the moduli space of K3 surfaces.

Jeff Viaclovsky Collapsing Ricci-flat metrics on K3 surfaces

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The model metric

Gibbons-Hawking ansatz over U = T 2

x,y × Rz>0, with

V = 2π A bz for a positive integer b > 0. Total space N has one complete end as z → ∞ and one incomplete end as z → 0. Choosing the connection form to be θb = (b/A)(dt − xdy), we can write gmodel = 2πbz A (dx2 + dy2 + dz2) + A 2πbz θ2

b,

with

• dθ = b

AdvolT 2,

• The level sets {z = constant} are identified with Nil3

b(ǫ, τ),

with a left-invariant metric (depending on z).

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Hyperk¨ ahler triples

The forms ω1 = dz ∧ θ + V dx ∧ dy ω2 = dx ∧ θ + V dy ∧ dz ω3 = dy ∧ θ + V dz ∧ dx. are a hyperk¨ ahler triple, ωi ∧ ωj = δijdvolg. We will need to construct an approximate hyperk¨ ahler triple on the “glued” manifold.

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ALHb metrics

Making the substitution z = (3/2)s2/3, and then scaling appropriately, the metric takes the form ds2 + s2/3gT 2 + s−2/3 A 3bπθb 2 .

• Volume growth is O(s4/3).
• Rm ∈ L2
• |Rm| = O(s−2) as s → ∞, but not any better. Thus these

asymptotics do not fall under the classification of Chen-Chen. If b = 0 and V = constant, this is ALH geometry. For b = 0, we will therefore refer to this type of geometry as ALHb geometry.

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ALHb ends

The red circles represent the S1 fibers, the blue curves represent the T 2s. Note that, in terms of distance to a basepoint, diam(T 2

s ) ∼ s1/3

diam(S1

s) ∼ s−1/3.

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Tian-Yau metrics are ALHb

Theorem (HSVZ) A Tian-Yau metric on Xb = DPb \ T 2 is ALHb, with g = gmodel,b + O(e−δs2/3) as s → ∞, for some δ > 0. The proof relies on finding good asymptotics for the complex structure, and then using techniques in Hein’s thesis and Tian-Yau. Gauge transformation to make the leading term of the connection

• ur standard choice.

Moreover, there is a hyperk¨ ahler triple which is asymptotic to our model hyperk¨ ahler triple.

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The neck potential

Choose p1, . . . , pb++b− ∈ T 2 × R. There exists V : T 2 × R \ P → R such that

• ∆V = 0
• V ∼ 1

2r near each monopole point.

• 1

2π ∗ dV ∈ H2(T 2 × R \ P, Z).

• V = O(e−δ|z|) +
• 2π

A b−z + c−

z ≪ 0 − 2π

A b+z + c+

z ≫ 0 Proof: in the universal cover, at large distances, V looks like electric potential of a collection of uniformly charged plates. Free to add kz to fix leading terms.

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The neck metric

Since

1 2π ∗ dV ∈ H2(T 2 × R \ P, Z). there is a corresponding

S1-bundle S1

N

π

• T 2 × R \ P.

and a connection form θ so that Ω = dθ = ∗dV. The neck metric: gN = V (gT 2 + dz2) + V −1θ2.

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Problem

Problem: Gibbons-Hawking requires a positive harmonic function, but the above electric potential is negative. Solution: add a large constant: Vβ = V + β, where β ≫ 0. This gives us an incomplete metric on the region N(T−, T+), where −T− < z < T+, where T± ∼ β. Analogous to Ooguri-Vafa metric, our case is a doubly-periodic analogue of this.

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Other key points of proof

• Analysis of harmonic functions on ALHb spaces, and Liouville

Theorems.

• Analysis of rescaled geometry of approximate metrics.
• Definition of weighted H¨
• lder spaces and weighted Schauder

estimate.

• Main blow-up analysis to prove uniform injectivity of linearized
• perator of hyperk¨

ahler triple gluing.

Jeff Viaclovsky Collapsing Ricci-flat metrics on K3 surfaces

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End

Thank you for your attention.

Jeff Viaclovsky Collapsing Ricci-flat metrics on K3 surfaces