j. differential geometry

55 (2000) 475-546

LARGE COMPLEX STRUCTURE LIMITS OF K3 SURFACES

MARK GROSS & P. M. H. WILSON

Abstract

Motivated by the picture of mirror symmetry suggested by Strominger, Yau and Zaslow, we make a conjecture concerning the Gromov-Hausdorff limits of Calabi-Yau n-folds (with Ricci-flat Kähler metric) as one approaches a large complex structure limit point in moduli; a similar conjecture was made independently by Kontsevich, Soibelman and Todorov. Roughly stated, the conjecture says that, if the metrics are normalized to have constant diameter, then this limit is the base of the conjectural special lagrangian torus fibrations associated with the large complex structure limit, namely an nsphere, and that the metric on this S^{n} is induced from a standard (singular) Riemannian metric on the base, the singularities of the metric corresponding to the limit discriminant locus of the fibrations. This conjecture is trivially true for elliptic curves; in this paper we prove it in the case of K3 surfaces. Using the standard description of mirror symmetry for K3 surfaces and the hyperkähler rotation trick, we reduce the problem to that of studying Kähler degenerations of elliptic K3 surfaces, with the Kähler class approaching the wall of the Kähler cone corresponding to the fibration and the volume normalized to be one. Here we are able to write down a remarkably accurate approximation to the Ricci-flat metric: if the elliptic fibres are of area ϵ > 0, then the error is O(e^{−C/ϵ}) for some constant C > 0. This metric is obtained by gluing together a semi-flat metric on the smooth part of the fibration with suitable Ooguri-Vafa metrics near the singular fibres. For small ϵ, this is a sufficiently good approximation that the above conjecture is then an easy consequence.

0. Introduction

The notion of large complex structure limit plays a special role in the theory of mirror symmetry. If X is a Calabi–Yau manifold, a large complex structure limit point is a point in a compactified moduli space

Received October 9, 2000, and, in revised form, April 3, 2001. The first author was supported in part by NSF grant DMS-9700761, Trinity College, Cambridge, and EPSRC.

of complex structures \overline{M}_{X} on X which, in some sense, represents the “worst possible degeneration” of the complex structure. This notion was given a precise Hodge-theoretic meaning in [27]. The basic example to keep in mind of this sort of degeneration is the degeneration of a hypersurface of degree n + 1 in P^{n} to a union of the n + 1 coordinate hyperplanes. Mirror symmetry posits the existence of a mirror to X associated to each large complex structure limit point of X. To first approximation, this means that if p ∈ \overline{M}_{X} is a large complex structure limit point in a compactification of the complex moduli space of X, then there exists a mirror X̌ and an isomorphism between a neighbourhood of p in \overline{M}_{X} and the complexified Kähler moduli space of X̌ which preserves certain additional information, such as the Yukawa couplings (which will not concern us in this paper). This isomorphism is known as the mirror map.

Now the Strominger-Yau-Zaslow conjecture [32] suggests that mirror symmetry can be explained by the existence of a special Lagrangian fibration on X when the complex structure on X is near a large complex structure limit point. The mirror X̌ is then expected to be constructed as the dual of this special Lagrangian fibration. The notion of special Lagrangian is a metric one: it depends on both the complex structure (determined by a holomorphic n-form Ω on X, where n = dim_{C} X), and a Ricci-flat Kähler metric, determined by its Kähler form ω. Thus we expect the existence of special Lagrangian fibrations will depend a great deal on the metric properties of Calabi–Yau manifolds near large complex structure limit points.

The simplest example of such a situation occurs for elliptic curves. Consider the family of elliptic curves E_{α} = C/⟨1, iα⟩, with α → ∞. We also choose a Ricci-flat, i.e., flat, metric g, which we will take to be the standard Euclidean metric. As α → ∞, the complex structure approaches the large complex structure limit point in the moduli space of elliptic curves; the period iα is approaching the cusp point of the compactification of H/SL_{2}(Z).

Now given the metric g, as α → ∞ it is clear that these elliptic curves converge to an infinitely long cylinder. However, if we rescale the metric, with g_{α} = g/α, then V ol(E_{α}) = 1 in this metric. With this metric, we can instead view E_{α} as C/⟨1/\sqrt{α}, i\sqrt{α}⟩ with the standard Euclidean metric, and then E_{α} converges to a line as α → ∞.

Finally, we may renormalize the metric again so that the diameter of E_{α} remains bounded, with the metric g_{α} = g/α^{2}. Then E_{α} can be identified with C/⟨1/α, i⟩ with the Euclidean metric, and E_{α} converges to a circle.

Of course, in this situation the special Lagrangian T^{1}-fibration on E_{α} is E_{α} → S^{1} obtained by projection onto the imaginary axis. So with the second and third choices of normalization, the special Lagrangian fibres collapse.

This is a rather trivial example, but forms a good basis for speculating about what might happen in higher dimensions. Intuitively, if we normalize the metric so as to keep the volume of the manifold bounded, we expect to see the fibres of the hypothetical special Lagrangian fibration contracting down to points; if furthermore we normalize so as to have bounded diameter, we expect the Calabi–Yau manifold to “converge” (in a sense we will make more explicit in §6) to a sphere of dimension n.

To test this picture, and to improve our understanding of Ricci-flat metrics, we have chosen to study the metric on K3 surfaces approaching large complex structure limit points. This is made easier by the fact that special Lagrangian fibrations are known to exist on K3 surfaces by a standard trick of performing a hyperkähler rotation of the complex structure, so that one reduces the problem of finding a special Lagrangian fibration to that of finding an elliptic fibration. Using this, we show in §1 that after performing this hyperkähler rotation, approaching a large complex structure limit is more or less the same as fixing the complex structure on a K3 elliptic fibration f : X → P^{1}, and letting the Kähler form ω on X vary in such a way that the area of the fibres approaches zero. Thus we ask the question: what does a Ricci-flat metric on an elliptic K3 surface look like when the area of the fibres is very small?

This is an interesting question even if one is not interested in mirror symmetry. In [1], M. Anderson studied degenerations of Ricci-flat metrics on K3 surfaces. If the volume of the surface is fixed and the diameter remains bounded, then the metrics converge to an orbifold metric (corresponding to degeneration to a K3 with rational double points). This picture of the moduli space of K3 surfaces with orbifold metric was originally studied in [21]. If the diameter is unbounded, Anderson proved collapsing must occur, but gave no more detailed information. The case under consideration in this paper can be considered to be the most extreme degeneration of metric. In particular, the orbifold case and the elliptic fibration case are the only Kähler degenerations, in which the complex structure of the K3 surface is held fixed. We will in fact consider a slightly more general situation, where the complex structure still varies to some extent. This is described more clearly in §1.

We assume the generic case, so that f has 24 singular fibres, each of Kodaira type I_{1} (a pinched torus). If X_{0} denotes the complement of these 24 singular fibres, then it is possible to write down a family of explicit Ricci-flat metrics which we refer to as semi-flat: these metrics are in fact flat when restricted to the fibres. The semi-flat metric was first introduced in [12]. There, it was used to get a first approximation to a complete Ricci-flat metric on the complement of a fibre of a rational elliptic surface. In [12], an arbitrary metric was then glued in to take care of the singular fibres so that techniques of [33], [34] could be applied to obtain a complete Ricci-flat metric on this manifold. While we follow this idea in spirit, we have here a new ingredient we can take advantage of. There is an explicit Ricci-flat metric defined in a neighbourhood of each singular fibre, first written down by Ooguri and Vafa in [29]. It is not semi-flat, but it in fact decays to a semi-flat metric exponentially. We can glue 24 copies of the Ooguri–Vafa metric in to the semi-flat metric, and thereby obtain a metric which is remarkably close to being Ricci-flat: in fact, the Ricci curvature is bounded in absolute value by O(e^{−C/ϵ}), where ϵ denotes the area of a fibre. Thus as ϵ → 0, the Ricci curvature of this glued metric approaches zero very rapidly.

We then follow standard techniques to show that the genuine Ricciflat metric representing the same Kähler class is very close to the glued metric, hence showing the explicit metric we constructed is a very good approximation to the genuine metric. We follow the proof of Kobayashi in [20], based on the original methods of Yau [35] — cf. also [7], [33], [34]. In [20] Kobayashi proves that near a Kummer surface, the Ricci-flat metric on a K3 surface is close to the flat orbifold metric on the Kummer surface. While the techniques are the same, it is perhaps surprising that they apply in our circumstances. Indeed, if the volume of the K3 surface is held fixed, then as ϵ → 0, the diameter of our metric approaches ∞. Thus the relevant Sobolev constant approaches zero, and so it will be important to control this precisely. It turns out that everything works because the starting glued metric is already extremely close to being Ricci-flat.

More explicitly, for Kähler classes [ω_{ϵ}] on X, where ϵ denotes the volume of a fibre of f, we construct a representative Kähler metric ω_{ϵ} with very small Ricci curvature. Yau’s proof [35] of the Calabi conjecture yields a solution u_{ϵ} to the equations

(ω_{ϵ} + i∂ ¯∂u_{ϵ})^{2} = e^{F_{ϵ}}ω_{ϵ}^{2}

∫_{X} u_{ϵ}ω_{ϵ}^{2} = 0

with F_{ϵ} = log (\frac{Ω∧Ω̄/2}{ω_{ϵ}^{2}}). The metric ω_{ϵ} + i∂ ¯∂u_{ϵ} is the desired Ricci-flat metric. We obtain a global C^{2}-estimate (Lemma 5.3), namely that for some positive constant C,

C^{−1}ω_{ϵ} ≤ ω_{ϵ} + i∂ ¯∂u_{ϵ} ≤ Cω_{ϵ}.

Moreover, the main theorem of the paper (Theorem 5.6) states that for any simply connected open set U ⊂ B whose closure is disjoint from the discriminant locus of f, and for any k ≥ 2, 0 < α < 1, there exist positive constants C_{1}, C_{2}, ϵ_{0} such that, for all ϵ < ϵ_{0},

∥u_{ϵ}∥_{C^{k,α}} ≤ C_{1}e^{−C_{2}/ϵ},

where the C^{k,α} norm is on the set f^{−1}(U). Thus, away from the singular fibres, ω_{ϵ} is a very good approximation to the actual Ricci-flat metric. See Theorem 5.6 for a more precise statement, which requires some care in the choice of the Kähler class [ω_{ϵ}].

The information obtained gives a clear picture of the metric behaviour as ϵ → 0. Using the above results, we prove the fibres are collapsing to points, and that away from the singular fibres, the metric approaches the semi-flat metric. In fact we will compute the Gromov– Hausdorff limit of a sequence of K3 surfaces with ϵ → 0 and the metrics renormalized so that the diameter remains bounded. This limit is indeed an S^{2}, but the metric on the S^{2} is singular at precisely 24 points corresponding to the singular fibres. See §6 for more precise statements. There, we state a conjecture, also made independently by Kontsevich, Soibelman, and Todorov, about the Gromov–Hausdorff limit of Calabi– Yau manifolds approaching large complex structure limit points. The above results prove this conjecture in the two dimensional case.

The structure of the paper is as follows. In §1 we briefly review mirror symmetry for K3 surfaces, so as to reduce the problem to one of understanding elliptic fibrations. In §§2 and 3, we introduce various ways of thinking about Ricci-flat metrics on elliptic fibrations, and then discuss required properties of the semi-flat and Ooguri–Vafa metrics. In §4, we build the glued metric. In §5, we run through the standard program to obtain estimates for Ricci-flat metrics, proving the main result of the paper, Theorem 5.6. Finally, in §6, we relate these results to Gromov–Hausdorff convergence, and speculate as to what kind of results in this direction might be expected and useful in higher dimensions.

Acknowledgements. We would like to thank G. Gibbons, N. Hitchin, N.C. Leung, M. Micallef, M. Singer, Y. Soibelman, and R. Thomas. The first author would especially like to thank S.-T. Yau and E. Zaslow, with whom he held extensive discussions about the Ooguri– Vafa metric in 1998.

1. Identification of large complex structure limits

There are a number of variants of mirror symmetry for K3 surfaces: see especially [10] for mirror symmetry between algebraic families of K3 surfaces and [4] for a more general version. We will use an intermediate version here, following [14], §7, which highlights the role of the special Lagrangian fibration. See also [17], §1. We review this point of view here. This will serve as motivation for Question 1.2 below, which will be addressed in the remainder of the paper. However, the setup of mirror symmetry will not be used again in this paper.

Let L be the K3 lattice, L = H^{2}(X, Z) for X a K3 surface. Fix a sublattice of L isomorphic to the hyperbolic plane H generated by E and σ_{0}, with E^{2} = 0, σ_{0}^{2} = −2, and E.σ_{0} = 1. We will view mirror symmetry as an involution acting on the moduli space of triples (X, B + iω, Ω) where X is a marked K3 surface, Ω is the class of a holomorphic 2-form on X, ω ∈ E^{⊥} ⊗ R a Kähler class on X, and the B-field B lies in E^{⊥}/E ⊗ R. In addition Ω is normalised so that Im Ω ∈ E^{⊥} ⊗ R and ω^{2} = (Re Ω)^{2} = (Im Ω)^{2}. Mirror symmetry interchanges (X, B + iω, Ω) with (X̌, B̌ + iω̌, Ω̌), where X̌ denotes a marked K3 surface with the following data:

Ω̌ ≡ (E. Re Ω)^{−1}(σ_{0} + B + iω) mod E
B̌ ≡ (E. Re Ω)^{−1} Re Ω − σ_{0} mod E
ω̌ ≡ (E. Re Ω)^{−1} Im Ω mod E.

The actual classes of Ω̌ and ω̌ are determined completely by the relations (Re Ω̌)^{2} = (Im Ω̌)^{2} = ω̌^{2} and ω̌.(Re Ω̌) = ω̌.(Im Ω̌) = (Re Ω̌).(Im Ω̌) = 0.

We can now identify the large complex structure limit of X̌. This limit is mirror to the large Kähler limit of X. In the latter limit, we keep the complex structure on X fixed but allow the Kähler form to go to infinity. More precisely, if {B_{l} + it_{l}ω} is a sequence of complexified Kähler forms on X with t_{l} > 0, t_{l} → ∞, then we say {B_{l} + it_{l}ω} are approaching the large Kähler limit in the complexified Kähler moduli space of X.

We will now take, for our purposes,

Definition 1.1. For each l, let X̌_{l} be the K3 surface given by the data (X̌_{l}, B̌_{l} + iω̌_{l}, Ω̌_{l}) mirror to (X, B_{l} + it_{l}ω, t_{l}Ω). The sequence of surfaces {X̌_{l}} is said to approach a large complex structure limit point.

We will take this as the starting point of our analysis, and will not prove here that this is equivalent to other reasonable definitions of large complex structure limits found in the literature (but see discussions in [10]).

The reader will note that we are cheating to some extent here, by only approaching the large Kähler limit along a ray. The more general approach might be to allow a more general sequence of Kähler forms. However, this is more difficult to deal with because the elliptic fibration which arises below will be varying. We will ignore this difficulty in this paper, as it obscures our main objectives.

Note that

Ω̌_{l} = (t_{l}E. Re Ω)^{−1}(σ_{0} + B_{l} + it_{l}ω) mod E

and

ω̌_{l} = (E. Re Ω)^{−1} Im Ω mod E.

More precisely, if a representative B_{l} for B_{l} mod E is chosen in E^{⊥} ⊗ R with the property that B_{l} · σ_{0} = 0, then the requirement that Ω̌_{l}^{2} = 0 yields

Ω̌_{l} = (t_{l}E. Re Ω)^{−1}(σ_{0}+(B_{l}+it_{l}ω)+(\frac{t_{l}^{2}ω^{2} − B_{l}^{2}}{2} +1−it_{l}ω.(σ_{0}+B_{l}))E).

Furthermore the requirement that ω̌_{l}.Ω̌_{l} = 0 yields

ω̌_{l} = (E. Re Ω)^{−1}(Im Ω − (Im Ω.(σ_{0} + B_{l}))E).

The Kähler class ω̌_{l} is represented by a Ricci-flat metric ǧ_{l}, and we would like to understand the behaviour of this metric as t_{l} → ∞. It is convenient to perform a hyperkähler rotation, i.e., ǧ_{l} is also a Kähler metric on the K3 surface X̌_{l,K} with

Ω̌_{l,K} = Im Ω̌_{l} + iω̌_{l}
ω̌_{l,K} = Re Ω̌_{l}.

This equality holds on the level of forms. Explicitly, in cohomology,

Ω̌_{l,K} = (E. Re Ω)^{−1}(ω + i Im Ω − ((ω + i Im Ω).(σ_{0} + B_{l}))E)
ω̌_{l,K} = (t_{l}E. Re Ω)^{−1}(σ_{0} + B_{l}) mod E.

We will assume that, for all l, E represents the class of a fibre of an elliptic fibration f_{l} : X̌_{l,K} → P^{1}. This elliptic fibration coincides with a special Lagrangian T^{2}-fibration on X̌_{l}. For general choice of data, such elliptic fibrations with fibre class E automatically exist, since then PicX̌_{l,K} = ZE and E^{2} = 0. For any choice of data, there always exists an elliptic fibration on X̌_{l,K}, but the class of the fibre might only be the image of E under reflections by −2 curves in PicX̌_{l,K}. (See [17], §1 for further details.)

Note that the area of the fibre of f_{l} under the metric ǧ_{l} is (t_{l}E. Re Ω)^{−1}, which goes to zero as t_{l} → ∞.

Now Ω̌_{l,K} depends on l, but these classes only differ by the pull-back of a class from P^{1}. This in fact tells us the elliptic K3 surfaces X̌_{l,K} are closely related. Indeed, if f : X → P^{1} is an elliptic K3 surface, with holomorphic 2-form Ω, then whenever α is a 2-form on P^{1}, Ω′ = Ω+f^{∗}α satisfies Ω′ ∧ Ω′ = 0 as forms, and thus Ω′ induces another complex structure on X such that f remains a holomorphic elliptic fibration in this new complex structure. All the surfaces X̌_{l,K} are clearly related in this way. In particular, all these elliptic surfaces have the same jacobian J̌_{K}, which is the unique elliptic K3 surface with a holomorphic section with complex structure induced by Ω̌_{l,K} + f_{l}^{∗}α for some α.

This now leads us to the following question:

Question 1.2. Let j : J → P^{1} be an elliptic K3 surface with a section, and let f_{l} : X_{l} → P^{1} be a sequence of elliptic K3 surfaces with jacobian j : J → P^{1}. Let ω_{l} be a Ricci-flat Kähler metric on X_{l} with V ol(X_{l}) independent of l. Let ϵ_{l} = Area_{ω_{l}}(f_{l}^{−1}(y)) for any point y ∈ P^{1}, and suppose ϵ_{l} → 0 as l → ∞. Describe the behaviour of the metric ω_{l} as l → ∞.

We will solve this question in this paper in the case that the map j has 24 Kodaira type I_{1} fibres. This is true for the generic K3 elliptic fibration.

We end this section with a few additional important comments about this setup.

First, it is often convenient to identify the underlying differentiable manifold of an elliptic K3 surface f : X → B with that of its jacobian. This can be done in a reasonably canonical fashion by choosing a C^{∞} section σ_{0} : B → X of f. If Ω_{X} is a holomorphic 2-form on X, then Ω_{J} = Ω_{X} − f^{∗}σ_{0}^{∗}Ω_{X} defines a new complex structure on X, in which σ_{0}(B) is a holomorphic section. This new complex structure yields the jacobian.

Another important point is that once a C^{∞} zero-section σ_{0} for f : X → B is chosen, we obtain a group structure on the non-singular part of each fibre of f. Let X^{0} ⊆ X be obtained by taking the union of the identity components of each fibre. Then given a holomorphic 2form Ω on X, we can construct a map from the holomorphic cotangent bundle T_{B}^{∗} to X^{0}, taking the zero section of T_{B}^{∗} to σ_{0}(B), and with the property that the pull-back of Ω to T_{B}^{∗} is a form Ω_{can} + α, where α is a 2-form pulled back from the base and Ω_{can} is the canonical holomorphic symplectic 2-form on T_{B}^{∗}. (See [14], §§2 and 7 for further details of this map.) The canonical holomorphic symplectic 2-form can be defined in local coordinates. If y is a local holomorphic coordinate on the base B, we can take x to be the corresponding canonical coordinate on the fibres of T_{B}^{∗}, so that the coordinate (x_{0}, y_{0}) represents the 1-form x_{0}dy at the point in B with coordinate y_{0}. The pair x, y are called holomorphic canonical coordinates. Then the canonical 2-form on T_{B}^{∗} is dx ∧ dy in these coordinates.

The map T_{B}^{∗} → X^{0} also gives an exact sequence

0 → R^{1}f_{∗}Z → T_{P^{1}}^{∗} → X^{0} → 0.

R^{1}f_{∗}Z gives a degenerating family of lattices in the fibres of the complex line bundle T_{B}^{∗}. Thus working on the cotangent bundle of B gives useful coordinates for X away from the singular fibres, and these coordinates will be used repeatedly in later sections.

2. Equations for Ricci-flatness

In this section we will discuss equations for Ricci-flatness in different coordinate systems. We are interested in the behaviour of the metric on an elliptic K3 fibration, and this metric behaves in radically different ways away from the singular fibres as opposed to a neighbourhood of the singular fibres. In these two different cases, it will be useful to have two different coordinate systems to study the metrics.

For studying the metric away from the singular fibres, we adopt the set-up from the previous section, with π : T_{B}^{∗} → B where B is an open subset of C. We are actually working on X = T_{B}^{∗}/Λ, where Λ is a holomorphically varying family of lattices in T_{B}^{∗}. We will assume in this section that the zero section is holomorphic, so that the holomorphic 2-form on X is induced by Ω = dx ∧ dy on T_{B}^{∗}, where y = y_{1} + iy_{2} and x = x_{1} + ix_{2} are holomorphic canonical coordinates on T_{B}^{∗}. The Kähler form in these coordinates takes the form

ω =\frac{i}{2}W (dx ∧ dx̄ + b̄ dx ∧ dȳ + b dy ∧ dx̄ + (W^{−2} + |b|^{2}) dy ∧ dȳ)
=\frac{i}{2}(W (dx + b dy) ∧ \overline{(dx + b dy)} + W^{−1} dy ∧ dȳ).

Here W and b are defined by the above expression, and the coefficient of dy ∧ dȳ is chosen to ensure the normalisation ω^{2} = (Im Ω)^{2}. The function W is real-valued and the function b is complex-valued. The Kähler condition is now dω = 0. This equation can be written as

∂_{y}W = ∂_{x}(W b)
∂_{y}(W b̄) = ∂_{x}(W (W^{−2} + |b|^{2})).

Note that expanding the second equation out gives

W ∂_{y}b̄ + b̄∂_{y}W = −W^{−2}∂_{x}W + (∂_{x}W)|b|^{2} + W (b ∂_{x}b̄ + b̄ ∂_{x}b).

Using the first equation to replace ∂_{y}W and simplifying gives the above two equations being equivalent to

(2.1)  (∂_{y} − b∂_{x})b̄ = −W^{−3}∂_{x}W
(2.2)  (∂_{y} − b∂_{x})W = W ∂_{x}b.

Define the vector fields

∂_{v} = W^{−1}∂_{x}
∂_{h} = ∂_{y} − b∂_{x}

and denote by ¯∂_{v} and ¯∂_{h} the complex conjugate vector fields. The subscripts v and h denote the vertical and horizontal vector fields respectively. Let ϑ_{v} and ϑ_{h} denote the dual frame of one-forms, i.e.,

ϑ_{v} = W (dx + bdy)
ϑ_{h} = dy.

Then

ω = \frac{i}{2}W^{−1}(ϑ_{v} ∧ ϑ̄_{v} + ϑ_{h} ∧ ϑ̄_{h}).

In addition, Equations (2.1) and (2.2) take the simpler form

(2.1′)  ∂_{h}b̄ = ∂_{v}W^{−1}

(2.2′)  −∂_{h}W^{−1} = ∂_{v}b.

Remark 2.1. While we don’t use this here, one can calculate that the holomorphic curvature Θ = (Θ_{ij})_{1≤i,j≤2} of this metric is given by

Θ_{11} = −Θ_{22} = ∂W ∧ ¯∂W^{−1} + W ∂ ¯∂W^{−1} + W^{2}∂b̄ ∧ ¯∂b
Θ_{21} = −Θ̄_{12} = −W^{−1}∂(W^{2} ¯∂b).

Example 2.2. The standard semi-flat metric. We call a metric semi-flat if it restricts to a flat metric on each elliptic fibre. As above, let B ⊆ C an open subset, y the coordinate on C. Let τ_{1}(y), τ_{2}(y) be two holomorphic functions on B such that τ_{1}(y)dy, τ_{2}(y)dy generate a lattice Λ(y) ⊆ T_{B,y}^{∗} for each y ∈ B, giving us the holomorphically varying family of lattices Λ ⊆ T_{B}^{∗} = B × C. Typically, we may allow τ_{1} and τ_{2} to be multi-valued. Assuming without loss of generality that Im(τ̄_{1}τ_{2}) > 0, then a Ricci-flat metric on X = (B × C)/Λ is given by the data

W = \frac{ϵ}{Im(τ̄_{1}τ_{2})}
b = −\frac{W}{ϵ} [Im(τ_{2}x̄)∂_{y}τ_{1} + Im(τ̄_{1}x)∂_{y}τ_{2}].

It is easy to check that these satisfy the equations (2.1) and (2.2). This metric, a priori defined on T_{B}^{∗}, descends to a metric on X, and the area of a fibre of f : X → B is ϵ. We call this metric on X the standard semi-flat metric, with Kähler form ω_{SF}.

The reader may check explicitly that this metric is independent of the particular choice of generators for Λ, so that multi-valuedness of τ_{1} and τ_{2} do not cause a problem. Furthermore, the metric is independent of the choice of the coordinate y (keeping in mind that a change of the coordinate y necessitates a change of the canonical coordinate x, and hence the functions τ_{1}, τ_{2}). This may also be seen as follows: The inclusion R^{1}f_{∗}Z ≅ Λ ⊆ T_{B}^{∗} allows one to identify (R^{1}f_{∗}R)⊗C^{∞}(B) with the underlying C^{∞} vector bundle T_{B}^{∗}, along with the Gauss-Manin connection ∇_{GM} on T_{B}^{∗}, the flat connection whose flat sections are sections of R^{1}f_{∗}R. The standard semi-flat metric is the unique semi-flat Ricci-flat Kähler metric satisfying the conditions:

(1) The area of each fibre is ϵ; (2) ω_{SF}^{2} = (Re Ω)^{2} = (Im Ω)^{2};

(3) The orthogonal complement of each vertical tangent space is the horizontal tangent space of ∇_{GM} at that point.

This metric was described in [12], and in the more general context of special Lagrangian fibrations in [19], as well as [14], Example 6.4.

The reader should be aware however that if T_{σ} : X → X denotes translation by a holomorphic section σ, then T_{σ}^{∗}ω_{SF} may give rise to a different semi-flat metric, satisfying conditions (1) and (2) but not (3). However, if σ is not only holomorphic but a flat section with respect to the Gauss-Manin connection (so that σ(y) = a_{1}τ_{1}(y) + a_{2}τ_{2}(y) for constants a_{1}, a_{2}) then T_{σ} is an isometry and T_{σ}^{∗}ω_{SF} = ω_{SF}, T_{σ}^{∗}Ω = Ω.

It will also be useful to have the Kähler potential for the metric. This is a function φ such that ω = \frac{i}{2}∂ ¯∂φ. Let ϕ_{1} and ϕ_{2} be anti-derivatives of τ_{1} and τ_{2} respectively. Then we can take

φ = \frac{ϵ}{Im(τ̄_{1}τ_{2})} (−\frac{x̄^{2}}{2} \frac{τ_{1}}{τ̄_{1}} + |x|^{2} − \frac{x^{2}}{2} \frac{τ̄_{1}}{τ_{1}} ) + \frac{i}{2ϵ}(ϕ_{1}ϕ̄_{2} − ϕ̄_{1}ϕ_{2}).

This is well-defined on subsets T_{U}^{∗} ⊆ T_{B}^{∗} for U simply connected, but not on T_{B}^{∗}/Λ.

Construction 2.3. The Gibbons–Hawking Ansatz. We now describe the system of coordinates which is most suited to studying the hyperkähler metric in a neighbourhood of a singular fibre of the elliptic fibration. This system of coordinates goes under the name of the Gibbons–Hawking Ansatz, and the description in terms of a connection form on an S^{1}-bundle explained below is essentially the same as that given in [2], which in turn is based on earlier work of Gibbons and Hawking, Hitchin, and others.

Let U ⊆ R^{3} be an open set with the Euclidean metric, with coordinates u_{1}, u_{2}, u_{3}. Let π : X → U be a principal S^{1} bundle, with S^{1} action S^{1} × X → X written as (e^{it}, x) ↦ e^{it} · x. Let θ be a connection 1-form on X, i.e., a 𝔲(1) = iR-valued 1-form invariant under the S^{1}-action and such that θ(∂/∂t) = i. The curvature of the connection θ is dθ = π^{∗}α for a 2-form α on U, and iα/2π represents the first Chern class of the bundle (see [8], Appendix). Suppose V is a positive real function on U satisfying ∗dV = α/2πi. Let

ω_{1} = du_{1} ∧ θ/2πi + V du_{2} ∧ du_{3}
ω_{2} = du_{2} ∧ θ/2πi + V du_{3} ∧ du_{1}
ω_{3} = du_{3} ∧ θ/2πi + V du_{1} ∧ du_{2}.

Then ω_{1}^{2} = ω_{2}^{2} = ω_{3}^{2} is nowhere zero, and ω_{i} ∧ ω_{j} = 0, for i ≠ j. Furthermore, ∗dV = α/2πi implies dω_{i} = 0 for all i, since for instance

dω_{1} = −du_{1}∧dθ/2πi+dV ∧du_{2}∧du_{3} = −du_{1}∧∗dV +dV ∧du_{2}∧du_{3} = 0.

Therefore ω_{1}, ω_{2}, ω_{3} define a hyperkähler metric on X. Note V is harmonic, since dα = 0 implies that ∗d ∗ dV = 0.

Let θ_{0} denote the real 1-form θ/2πi, and observe that

−ω_{1} − iω_{2} = (θ_{0} − iV du_{3}) ∧ (du_{1} + idu_{2}).

By taking this to be the (holomorphic) 2-form Ω on X, this determines an integrable almost complex structure on X, where du_{1} + idu_{2} and θ_{0} − iV du_{3} span the holomorphic cotangent space inside the complexified cotangent space. It follows that the (integrable) almost complex structure J on the cotangent space is given by

J(du_{1}) = −du_{2}, J(du_{3}) = −V^{−1}θ_{0}.

Thus, if we consider the Kähler form ω = ω_{3} as an alternating tensor, and use the relation that if g is the Riemannian metric, then g(ζ, ξ) = ω(ζ, Jξ), we obtain an expression for the metric

ds^{2} = V du · du + V^{−1}θ_{0}^{2}.

Usually, we shall in fact start from a positive harmonic function V on U such that − ∗ dV represents the Chern class of the bundle. Then we can always find a connection 1-form θ with dθ/2πi = ∗dV, such a θ being uniquely determined up to pull-backs of closed 1-forms from U, and hence we obtain hyperkähler metrics as above.

Remark 2.4. We will need to calculate some information about the curvature of this metric. We can work locally, and therefore take the orthonormal moving coframe given by V^{1/2}du_{1}, V^{1/2}du_{2}, V^{1/2}du_{3} and V^{−1/2}θ_{0}. We can moreover write the connection form locally as

θ_{0} = \frac{dt}{2π} + A_{1}du_{1} + A_{2}du_{2} + A_{3}du_{3},

where ∇V = ∇ × A. To calculate the curvature, we may then apply Cartan’s method. We obtain

∥R∥^{2} = 12V^{−6}|∇V |^{4} + V^{−4}Δ(|∇V |^{2}) − 6V^{−5}(∇V) · (∇(|∇V |^{2})).

Using the fact that V is harmonic, we then recover the compact formula given in Equation (32) of [28] that

∥R∥^{2} = \frac{1}{2}V^{−1}ΔΔ(V^{−1}).

Example 2.5. If we consider the natural map C^{2} \ (0, 0) → P^{1}(C) = S^{2}, and restrict to S^{3} ⊂ C^{2}, we easily check that the image of (z_{1}, z_{2}) ∈ S^{3} is

(2 Re(z_{1}z̄_{2}), 2 Im(z_{1}z̄_{2}), |z_{1}|^{2} − |z_{2}|^{2}).

This is the standard Poincaré map. The formula also defines a map X = C^{2} \ (0, 0) → R^{3} \ (0, 0, 0); we compose this map with complex conjugation on z_{2} to obtain a map p : X = C^{2} \ (0, 0) → R^{3} \ (0, 0, 0), given by

p(z_{1}, z_{2}) = (2 Re(z_{1}z_{2}), 2 Im(z_{1}z_{2}), |z_{1}|^{2} − |z_{2}|^{2}).

This map exhibits X as an S^{1}-bundle over R^{3} \ (0, 0, 0), with Chern class ±1. The action of S^{1} on X is given by e^{it} ·(z_{1}, z_{2}) = (e^{it}z_{1}, e^{−it}z_{2}). Note also that if we compose p with projection onto the first two factors, we obtain the map sending (z_{1}, z_{2}) to 2z_{1}z_{2}, holomorphic with respect to the standard complex structures.

We now choose a positive harmonic function V on R^{3} \ (0, 0, 0) such that

− ∫_{S^{2}} ∗dV = ∫_{S^{2}} iα/2π = ±1,

i.e., the Chern number is correct. The particular examples of such V we consider are

V = e + \frac{1}{4π|u|} = e + \frac{1}{4π\sqrt{u_{1}^{2} + u_{2}^{2} + u_{3}^{2}}} ,

where e ≥ 0. The integral

∫_{S^{2}} ∗d (\frac{1}{4π\sqrt{u_{1}^{2} + u_{2}^{2} + u_{3}^{2}}} )

is easily seen to be ±1 (depending on the orientation of the sphere). Now we take as connection form

θ = iIm(z̄_{1}dz_{1} − z̄_{2}dz_{2})/(|z_{1}|^{2} + |z_{2}|^{2}).

Then

dθ/2πi = \frac{−(u_{1}du_{2} ∧ du_{3} + u_{2}du_{3} ∧ du_{1} + u_{3}du_{1} ∧ du_{2})}{4π(u_{1}^{2} + u_{2}^{2} + u_{3}^{2})^{3/2}} = ∗dV

as required. We therefore obtain hyperkähler metrics on X, which, for all e ≥ 0, extend to metrics on C^{2}. In fact, such metrics are ALF (asymptotically locally flat), approaching a flat metric when |u| → ∞, whilst being periodic in t. When e = 1, the metric obtained is the Taub-NUT metric, and when e = 0, it is just a flat metric on C^{2}. To prove the assertions for e = 0, straightforward calculations show that, with z_{j} = x_{j} + iy_{j},

ω_{1} = \frac{1}{π} (dx_{2} ∧ dy_{1} − dx_{1} ∧ dy_{2})
ω_{2} = \frac{1}{π} (dx_{1} ∧ dx_{2} − dy_{1} ∧ dy_{2})
ω_{3} = \frac{1}{π} (dx_{1} ∧ dy_{1} + dx_{2} ∧ dy_{2}).

So ω_{1}, ω_{2}, ω_{3} extend to C^{2}, and yield a flat metric, as claimed.

Construction 2.6. Gibbons–Hawking versus holomorphic coordinates. In the Gibbons–Hawking Ansatz, we consider the case when U = B × R, with B a contractible open subset of R^{2} — in particular, the S^{1}-bundle X over U is topologically trivial. Set y_{1} = u_{1}, y_{2} = u_{2}, so then y = y_{1} + iy_{2} is a complex coordinate on B. We will see below how the hyperkähler structure on X gives rise to a complex structure on X under which the function y is holomorphic, i.e., the map X → B is holomorphic. Moreover, if we pass to the universal cover X̃ on X, we can construct a holomorphic coordinate x (depending on a choice of holomorphic section of X̃ over B) such that the holomorphic 2-form is just dx ∧ dy. This in turn enables us to identify X̃ with T_{B}^{∗} over B, with x, y then being holomorphic canonical coordinates on T_{B}^{∗}, where the identification depends on our choice of holomorphic section. The S^{1}-action on X yields an R-action on T_{B}^{∗}, which we shall see is just translation on x_{1} = Re x, and so X is isomorphic to T_{B}^{∗}/Z. The Kähler form provided by the Gibbons–Hawking Ansatz yields a Kähler form ω on T_{B}^{∗}, corresponding of course to a Ricci-flat metric, and for which the functions W and b are independent of x_{1}. The Kähler form therefore descends to T_{B}^{∗}/Z, and is invariant under the obvious S^{1}-action.

Conversely, we shall see that any Ricci flat, S^{1}-invariant Kähler structure on T_{B}^{∗}/Z of the above type (i.e., we have x, y holomorphic canonical coordinates on T_{B}^{∗} over B, for which W and b are independent of x_{1}) does in fact arise from the Gibbons–Hawking Ansatz in the way that has just been described. Moreover, Gibbons–Hawking coordinates u_{1}, u_{2}, u_{3} and the connection form θ on X may be recovered from the holomorphic canonical coordinates x, y on T_{B}^{∗}. Here we have u_{1} = y_{1}, u_{2} = y_{2}, and u_{3} determined up to a constant.

We now give the details for the construction. We have U = B × R, with B a contractible open subset of R^{2}, and we set y = y_{1} + iy_{2}, a complex coordinate on B. Then dy_{1}+idy_{2} = dy, and from the Gibbons– Hawking Ansatz equations we observe that dy ∧ d(θ_{0} − iV du_{3}) = 0. By the theorem on integrability of almost complex structures, Ω = (θ_{0} − iV du_{3}) ∧ dy is a holomorphic 2-form for an integrable almost complex structure J on X, and locally there exists a holomorphic coordinate z such that dz = (θ_{0} − iV du_{3}) mod dy. Moreover it is then clear that z is determined up to a holomorphic function of y, and that locally the holomorphic coordinates recover the (integrable) complex structure. We now pass to the universal cover X̃ of X, topologically B × R^{2}, together with its integrable complex structure J̃ obtained from J (from now on, we shall work on X̃, but omit tildes from forms and functions pulled back from X). We note that the complex structure is invariant under the R-action on X̃ induced from the given S^{1}-action on X. The (global) form θ_{0} − iV du_{3} restricts down to a holomorphic 1-form on each fibre, locally just dz. Therefore, by integrating θ_{0} − iV du_{3} along paths in the fibre from some fixed point, we obtain a holomorphic coordinate on the fibre, which locally (up to a constant depending on the choice of base point) will coincide with z.

In order to get a global holomorphic coordinate x on X̃, we choose a holomorphic section of X̃ over B (such sections always exist), which will then be regarded as giving the required base point in each fibre for the path integration. In this way, we obtain a global holomorphic function x on X̃ such that x, y are holomorphic coordinates everywhere, and where x is uniquely determined up to a holomorphic function of y (corresponding to the choice of holomorphic section). By construction, the global holomorphic coordinates x, y on X̃ realize the almost complex structure, with y a holomorphic coordinate on the base and x a holomorphic coordinate on the fibres. Moreover Ω = −ω_{1} − iω_{2} = dx ∧ dy, and so we can identify X̃ → B with T_{B}^{∗} → B (with holomorphic canonical coordinates, as described in §1), where the chosen holomorphic section of X̃ over B is identified the zero section of the holomorphic cotangent bundle. Choosing a section of X̃ over U = B × R enables us to consider the coordinate t on S^{1} as a coordinate on the fibres; the above derivation of the holomorphic coordinate x then shows that its real part x_{1} = \frac{t}{2π} + g(y_{1}, y_{2}, u_{3}), for some function g, and that the action of R on X̃ is the obvious one given by translation on x_{1}. Explicitly X is obtained as a quotient of X̃ under the action of Z given by x_{1} ↦ x_{1} + 1.

Since dx = θ_{0} − iV du_{3} mod dy, there exists a complex-valued function b on X̃ such that dx + bdy = θ_{0} − iV du_{3}. Also

(dx + bdy) ∧ \overline{(dx + bdy)} = 2iV θ_{0} ∧ du_{3}.

We now set W = V^{−1} and calculate the Kähler form ω_{3} in terms of the holomorphic coordinates:

ω_{3} = du_{3}∧θ_{0}+V du_{1}∧du_{2} = \frac{i}{2}(W (dx+bdy)∧\overline{(dx + bdy)}+W^{−1}dy∧dȳ),

which we observe has the same form as our original general formula for ω in holomorphic canonical coordinates. Since we started with a Ricci-flat metric, the previous equations for Ricci-flatness (2.1) and (2.2) which we derived are then automatically satisfied.

The next point is to observe that W and b are independent of x_{1}, the real part of x. To see this, recall now that

ϑ_{v} = W (dx + bdy) = (V^{−1}θ_{0} − idu_{3})

and ϑ_{h} = dy is a globally defined coframe for the holomorphic cotangent bundle of X̃. In particular, since the imaginary part of ϑ_{v} is −du_{3}, we will have that the imaginary part of dϑ_{v} is zero. We first calculate

(2.3) ∂ϑ_{v} = W^{−1}∂W ∧ ϑ_{v} + W ∂b ∧ ϑ_{h} = (W^{−1}∂_{h}W − W ∂_{v}b) ϑ_{h} ∧ ϑ_{v}.

From this it is seen that Equation (2.2′) is just the statement that ∂ϑ_{v} = 0. We next calculate

¯∂ϑ_{v} = W^{−1} ¯∂W ∧ ϑ_{v} + W ¯∂b ∧ ϑ_{h}
= W^{−1} ¯∂_{v}W ϑ̄_{v} ∧ ϑ_{v} + W^{−1} ¯∂_{h}W ϑ̄_{h} ∧ ϑ_{v}
+ W ¯∂_{v}b ϑ̄_{v} ∧ ϑ_{h} + W ¯∂_{h}b ϑ̄_{h} ∧ ϑ_{h}.

(2.4)

If then Equation (2.1′) also holds, it is easily checked that the imaginary part of ¯∂ϑ_{v} is zero if and only if ∂_{v}W = −¯∂_{v}W and ∂_{v}b = −¯∂_{v}b, that is W and b are independent of x_{1}, the real part of x. Thus b is invariant under the R-action, that is b is the pull-back of a function from U.

Conversely, if we start from a Ricci-flat, S^{1}-invariant Kähler metric on T_{B}^{∗}/Z of the above type (i.e., we have x, y holomorphic canonical coordinates on T_{B}^{∗} over B, for which W and b are independent of x_{1}), we can pass to the universal cover X̃ = T_{B}^{∗} over B. The above construction then reverses. We set ϕ to be the imaginary part of ϑ_{v} = W (dx + bdy). Clearly ϕ is invariant under the given R-action on X̃. Reversing the derivation of the previous paragraph ensures that dϕ = 0 on X̃, and so there is a global function u_{3} with ϕ = −du_{3}, where u_{3} is invariant under the action of R, and is determined up to a constant. We set V = W^{−1}, θ_{0} = (dx + bdy) − iV ϕ and θ = 2πiθ_{0}; thus both V and θ are also invariant under the action of R. It is straightforward now to verify that we get back the above form of the Gibbons–Hawking Ansatz, with u_{1} = y_{1} and u_{2} = y_{2}, and where U = B × R is the quotient of X̃ by the R-action. The periodicity of this R-action then yields an S^{1}-bundle X over U (to which V and θ descend, and on which the corresponding S^{1}-action leaves V and θ invariant).

Finally, we calculate (for use in §4) the differential p_{∗}, where p : X̃ → U = B × R is the natural projection. Using the expression du_{3} = \frac{i}{2}W (dx − dx̄) + \frac{i}{2}W (bdy − b̄dȳ), we obtain

\begin{aligned}p_{∗}∂_{x} & = \frac{iW}{2} ∂_{u_{3}} \\ p_{∗}∂x̄ & = \frac{−iW}{2} ∂_{u_{3}} \\ p_{∗}∂_{y} & = ∂_{y} + \frac{iW}{2} b∂_{u_{3}} \\ p_{∗}∂ȳ & = ∂ȳ − \frac{iW}{2} b̄∂_{u_{3}}. \end{aligned} (2.5)

Thus p_{∗}∂_{h} = ∂_{y}, p_{∗}∂_{v} = \frac{i}{2}∂_{u_{3}}.

Also, as W and b can be thought of as functions on B × R, being independent of x_{1}, the formula (2.2′) translates into

(2.6)  −∂_{y}V = \frac{i}{2}∂_{u_{3}}b.

Thus b can be calculated as

(2.7)  b(y, u_{3}) = σ(y) + ∫ 2i∂_{y}V du_{3}

where σ(y) is some constant of integration.

S^{1}-invariant Ricci flat metrics on elliptic fibrations

We shall be most interested in the transformation described above when V and θ are themselves periodic in u = u_{3}. The hyperkähler metric descends to one on the corresponding S^{1}-fibration over Y = B × S^{1} if and only if the three 2-forms ω_{1}, ω_{2}, ω_{3} are invariant under changing u by a period, which in turn is saying that the periodicity in u is independent of y. We shall now change notation and denote this S^{1} × S^{1} fibration over B by X (the universal cover X̃ being the same as before). Since the restriction of the Kähler form ω_{3} to a fibre X_{y} is just du ∧ θ_{0} = du ∧ dt/2π, the volume of any fibre is just the periodicity in u. Changing coordinates to the holomorphic coordinates of Construction 2.6, we obtain a holomorphic map f : X → B to a contractible open subset B of C, whose fibres are elliptic curves. Having chosen a holomorphic section, we obtain holomorphic canonical coordinates x, y on the corresponding line bundle X̃ over B, where the holomorphic 2form Ω = dx ∧ dy, and where the Kähler form ω (as defined by the usual formula) determines a hyperkähler metric on X. Moreover, both W and b are independent of x_{1}.

The periods of the above elliptic fibration have a basis {1, τ(y)}, for some holomorphic function τ of y. If we wish to have an explicit formula for τ(y), we take a basis of homology {γ_{1}, γ_{2}}, where γ_{1} is an S^{1} in a fibre X_{y} of X → B given by the orbit of the S^{1}-action, and γ_{2} is an S^{1} in X_{y} mapping isomorphically to {y} × S^{1} ⊂ Y. Restricted to the fibre X_{y}, we have dx = θ_{0} − iV du_{3}; one of the periods is then

∫_{γ_{1}} dx = ∫_{γ_{1}} θ_{0} = 1,

as already observed, whilst the other period

τ(y) = ∫_{γ_{2}} dx = ∫_{γ_{2}} θ_{0} − i ∫_{γ_{2}} V du_{3}.

By choosing the appropriate orientation for γ_{2}, we may also assume that Im τ(y) > 0.

If we have such a holomorphic elliptic fibration f : X → B and Ricciflat metric (independent of x_{1}), we shall refer to it as an S^{1}-invariant Ricci-flat metric (on X) in canonical form.

Conversely, if we are given such an S^{1}-invariant Ricci-flat metric on X, we saw above how this does indeed arise from the Gibbons–Hawking Ansatz. Moreover, in this case, we also have that V and θ are periodic in u, with the period in u being constant, namely the volume of the elliptic fibres of f : X → B.

Remark 2.7. A particular case of an S^{1}-invariant Ricci-flat metric in canonical form is a semi-flat metric: Given, locally, two periods τ_{1} and τ_{2}, these should be interpreted as 1-forms on B, i.e., are τ_{1}dy, τ_{2}dy. We can then locally replace y with a holomorphic function g on an open set U such that dg = τ_{1}dy, and thus can assume τ_{1} = 1. Then in these coordinates, the semi-flat metric coincides with the Gibbons–Hawking metric obtained by taking V = Im τ_{2}/ϵ on U × R/ϵZ. We can then use the formula of Remark 2.4 to compute ∥R∥^{2} for a semi-flat metric (which will coincide with the value calculated via Remark 2.1). Thus

∥R∥^{2} = \frac{1}{2}V^{−1}ΔΔV^{−1} = \frac{ϵ^{2}}{2} (Im τ_{2})^{−1}ΔΔ(Im τ_{2})^{−1}.

In particular, ∥R∥^{2} → 0 as ϵ → 0.

Returning now to the set-up in Question 1.2; away from the singular fibres, we expect that, as the volume ϵ of the fibres tends to zero, the metric (suitably normalized) will approach a semi-flat one. This expectation is motivated by the following result, which proves a slightly weaker version of the expected convergence for the S^{1}-invariant Ricciflat case, purely by local considerations, as a consequence of Harnack’s inequality for harmonic functions. Whilst we don’t expect a purely local proof of convergence in general (i.e., not assuming the S^{1}-invariance of the metrics), the main result of this paper (Theorem 5.6) will prove a very strong form of the expected convergence to a semi-flat metric (locally over the base) by means of global methods.

Proposition 2.8. Let π : X → B be an elliptic fibration with periods {1, τ(y)}, over the open disc B of radius R in C, with Im τ(y) > 0, and let B_{0} ⊂ B denote a smaller disc of radius R_{0} < R. Suppose we have a sequence on X of S^{1}-invariant Ricci-flat metrics g_{i} in canonical form (and with constant volume form), for which the volume ϵ_{i} := ϵ(g_{i}) of the fibres tends to zero as i → ∞. Then on π^{−1}(B_{0}) we have W_{i} := W (g_{i}) → 0 uniformly as i → ∞. On a fixed fibre, with periods {1, τ}, we have the stronger statement that ϵ_{i}^{−1}W_{i} Im τ → 1 uniformly as i → ∞.

Proof. Our assumption that the volume form is constant ensures that we can fix the holomorphic canonical coordinates x, y and the holomorphic 2-form Ω = dx ∧ dy independent of i. We now transform the coordinates to Gibbons–Hawking coordinates; the first claim is equivalent to W_{i} → 0 uniformly on B_{0} × R. If the volume of the fibres is ϵ_{i}, then the periodicity in u is ϵ_{i}. Fix R_{1} with R_{0} < R_{1} < R; then for ϵ_{i} << 1, the ball B̃_{1} in U = B × R with centre the origin and radius R_{1} will contain the set B_{0} × [0, ϵ_{i}], so it will suffice to show that W_{i} → 0 uniformly on B̃_{1} as i → ∞. Fix i for the moment so that ϵ_{i} is sufficiently small as above, and drop the subscript for convenience. Let B̃ denote the ball of radius R, with centre the origin. Recall now that W = V^{−1}. Given that V is harmonic on B̃, this is precisely the situation in which we can apply the strong form of Harnack’s inequality, as stated in Problem 2.6 on page 29 of [11], namely that for any point P ∈ B̃_{1},

\frac{(1 − R_{1}/R)}{(1 + R_{1}/R)^{2}} ≤ \frac{V (P)}{V (0)} ≤ \frac{(1 + R_{1}/R)}{(1 − R_{1}/R)^{2}} .

Thus, for P ∈ π^{−1}(B_{0}), the ratio W (P)/W (0) is bounded above and below by appropriate positive constants. For each y ∈ B_{0}, we can calculate the volume of the fibre X_{y} as

ϵ = ∫_{X_{y}} W dx_{1} ∧ dx_{2} = ∫_{0}^{Im τ(y)} W dx_{2}.

For R_{0} fixed and for y ∈ B_{0}, we also have that Im τ(y) is bounded above and below by appropriate positive constants. On any fibre X_{y} with y ∈ B_{0}, we can find a point at which W takes the average value on the fibre, namely ϵ/Im τ(y). Putting all these facts together yields the claim that W_{i} → 0 uniformly on B̃_{1} as i → ∞.

For the stronger statement on a fixed fibre, we can assume that the fibre is X_{0}, and that W takes the average value ϵ/Im τ at the centre 0 of the ball B̃. If we take a concentric ball B̃(r) of small radius r, it will still contain all of {0} × [0, ϵ], provided ϵ < r. Harnack’s inequality then yields

\frac{(1 − r/R)}{(1 + r/R)^{2}} ≤ \frac{W (0)}{W (P)} ≤ \frac{(1 + r/R)}{(1 − r/R)^{2}}

for all P ∈ X_{0}. By taking r arbitrarily small, these upper and lower bounds are arbitrarily close to 1, and hence ϵ_{i}^{−1}W_{i} Im τ → 1 uniformly on X_{0}. q.e.d.

3. The Ooguri–Vafa metric

The aim of this section is to describe a certain hyperkähler metric on a neighbourhood of each singular fibre in our elliptically fibred K3 surface, and to derive various estimates associated with this metric. If the fibres are assumed to have volume ϵ, then away from the singular fibre, this metric decays very rapidly, for small ϵ, to a semi-flat metric. We shall assume throughout that we only have singular fibres of Kodaira type I_{1}, and so locally around the singular fibre, one of the periods is invariant under monodromy (and in fact, by an appropriate choice of holomorphic coordinate y on the base, may be taken to be constant, value 1), whilst the other period will be multivalued and tend to infinity. The metric we define will be an S^{1}-invariant metric (as described in the previous section) on the smooth part of the fibration, and will be most conveniently described in the Gibbons–Hawking coordinates.

The metric we describe was first written down (in a slightly different form) by Ooguri and Vafa [29], and so will be referred to as the Ooguri– Vafa metric. In §4, we shall start with an Ooguri–Vafa metric in a neighbourhood of each singular fibre; by appropriately twisting these metrics, we’ll show that they may be glued with a semi–flat metric away from the singular fibres, hence obtaining a global metric, which is Ricci-flat away from the gluing regions, and which represents the correct Kähler class. For small ϵ, it is these metrics which approximate very accurately the global Ricci flat metric with the given Kähler class.

Before launching into the technical details, we shall briefly describe the basic idea behind the construction of the Ooguri–Vafa metric, which, given the description of the Gibbons–Hawking Ansatz in §2, should strike the reader as very natural. The harmonic function V we use will be periodic in u of period ϵ (the volume of the fibres), but have Taub–NUT type singularities on the fibre y = 0 at the points u ∈ ϵZ.

We take U = D × R \ {0} × ϵZ, or more precisely its quotient by ϵZ, where D ⊂ C is an open disc centred at the origin. We denote by y_{1}, y_{2} the coordinates on D ⊂ C, and by u the coordinate on R. We want to write down V harmonic on U, periodic in u and with singularities of the correct type at the points {0} × ϵZ. For instance, around zero, V should behave like a harmonic function plus a term \frac{1}{4π|x|}, from which it will follow that the total space X of the S^{1}-fibration over U extends (by adding a single point) to a manifold X̄ mapping onto Ū = D × R/ϵZ. In addition, the hyperkähler metric extends to X̄. We are led therefore to take V = V_{0} + f(y_{1}, y_{2}), where f is a harmonic function in y_{1}, y_{2} on D, and

V_{0} = \frac{1}{4π} ∑_{−∞}^{∞} (\frac{1}{\sqrt{(u + nϵ)^{2} + y_{1}^{2} + y_{2}^{2}}} − a_{|n|}) ,

where a_{n} = \frac{1}{nϵ} (n > 0), thus ensuring appropriate convergence, and a_{0} is chosen appropriately to ensure that the periods do not change as we change ϵ — that is, we are defining metrics on a fixed elliptic fibration. This choice of a_{0} also ensures, on a fixed annulus in D, that ϵV_{0} ∼ −\frac{1}{2π} log r as ϵ → 0, where r^{2} = y_{1}^{2} + y_{2}^{2}. In general, the periods around an I_{1} fibre may be assumed to be 1 and τ(y) = \frac{1}{2πi} log y + ih(y), where h is holomorphic in y = y_{1} + iy_{2}, and these may be achieved in our construction by taking V = V_{0} + f(y_{1}, y_{2}), where f denotes the real part of h.

We now give the technical details.

Lemma 3.1. Let

T_{j} = \frac{1}{4π} ∑_{n=−j}^{j} (\frac{1}{\sqrt{(u + nϵ)^{2} + y_{1}^{2} + y_{2}^{2}}} − a_{|n|})

where

a_{n} = \begin{cases}1/nϵ & n ≠ 0 \\ 2(−γ + log(2ϵ))/ϵ & n = 0 \end{cases}

and γ is Euler’s constant. Then:

(a) The sequence {T_{j}} converges uniformly on compact sets in D × R − {0} × ϵZ to a harmonic function V_{0}. Here D ⊆ C is the unit disc centred at the origin.

(b) V_{0} has an expansion, valid when |y| ≠ 0,

V_{0} = −\frac{1}{4πϵ} log |y|^{2} + ∑_{{m=−∞ \atop m≠0}}^{m=∞} \frac{1}{2πϵ}e^{2πimu/ϵ}K_{0}(2π|my|/ϵ)

where y = y_{1} + iy_{2} and K_{0} is the modified Bessel function. (See [3], p. 374.)

(c) There exists a constant C such that for any 0 < r_{0} < 1, there exists an ϵ_{0} > 0 such that for all ϵ < ϵ_{0}, |y| > r_{0},

|V_{0} + \frac{1}{4πϵ} log |y|^{2}| ≤ \frac{C}{ϵ} e^{−2π|y|/ϵ}.

(d) If r ≤ 1, and f is a harmonic function on the disc D_{r} of radius r such that f(y) − \frac{1}{4π} log |y|^{2} > 0 for |y| ≤ r, then there exists an ϵ_{0} such that for all ϵ < ϵ_{0},

V_{0} + f(y)/ϵ > 0

in D_{r} × R.

Proof. (a) Let p be the smallest integer greater than ϵ^{−1}\sqrt{1 + ϵ^{2}}. Then for 0 ≤ u ≤ ϵ, y_{1}^{2} + y_{2}^{2} ≤ 1, we have

\frac{1}{\sqrt{(u + nϵ)^{2} + y_{1}^{2} + y_{2}^{2}}} > a_{|n|+p}

for all n. Let

R_{j} = \frac{1}{4π} ∑_{n=−j}^{j} (\frac{1}{\sqrt{(u + nϵ)^{2} + y_{1}^{2} + y_{2}^{2}}} − a_{|n|+p}) .

Then for j > 2p,

T_{j} − R_{j} = \frac{1}{4π} (−a_{p} − a_{0} − 2 ∑_{n=1}^{p−1} a_{n} + 2 ∑_{n=j−p+1}^{j} a_{n+p}) .

Put C(ϵ) = \frac{1}{4π} (−a_{p} − a_{0} − 2 ∑_{n=1}^{p−1} a_{n}). Note that ∑_{n=j−p+1}^{j} a_{n+p} → 0 as j → ∞, so if R_{j} converges uniformly on compact sets to a harmonic function R, then T_{j} converges to a harmonic function R + C(ϵ). Now for 0 < u < ϵ, y_{1}^{2} + y_{2}^{2} < 1, R_{j} is a monotonically increasing sequence of harmonic functions (since all terms are positive). Furthermore, it is easy to check that, say, the sequence R_{j} is bounded at u = ϵ/2, y_{1} = y_{2} = 0. Thus by the Harnack convergence theorem, (Theorem 2.9, [11]), the R_{j} converge uniformly on compact subsets to a harmonic function R, and V_{0} = R + C(ϵ). Since R is positive, we see V_{0} > C(ϵ). For u = 0, ϵ, we merely omit the term which blows up and then repeat the previous argument.

(b) The part which requires care is the constant term of the Fourier expansion, i.e., computing \frac{1}{ϵ} ∫_{0}^{ϵ} V_{0}du. To do so, consider the following variant on the T_{j}:

S_{j} = \frac{1}{4π} ∑_{n=−j}^{j} (\frac{1}{\sqrt{(u + nϵ)^{2} + y_{1}^{2} + y_{2}^{2}}} − b_{|n|})

large complex structure limits 499

where

b_{n} = \begin{cases}(log(n + 1) − log n)/ϵ & n ≠ 0 \\ 0 & n = 0. \end{cases}

Then

T_{j} − S_{j} = \frac{1}{4π} (\frac{2}{ϵ} log(j + 1) − a_{0} − \frac{2}{ϵ} ∑_{n=1}^{j} \frac{1}{n}) .

As j → ∞, this converges to

\frac{1}{4π} (−\frac{2γ}{ϵ} − a_{0}) = −\frac{1}{2πϵ} log(2ϵ).

Now we calculate

4π ∫_{0}^{ϵ} S_{j}du = ∑_{n=−j}^{j} ∫_{0}^{ϵ} (\frac{1}{\sqrt{(u + nϵ)^{2} + |y|^{2}}} − b_{|n|}) du
= ∑_{{n=−j \atop n≠0}}^{j} (− log(|n| + 1) + log |n|)

+ ∑_{n=−j}^{j} ∫_{nϵ}^{(n+1)ϵ} \frac{1}{\sqrt{u^{2} + |y|^{2}}} du

= ∫_{ϵ}^{(j+1)ϵ} (\frac{1}{\sqrt{u^{2} + |y|^{2}}} − \frac{1}{u}) du

+ ∫_{−jϵ}^{0} (\frac{1}{\sqrt{u^{2} + |y|^{2}}} + \frac{1}{u − ϵ}) du

+ ∫_{0}^{ϵ} \frac{1}{\sqrt{u^{2} + |y|^{2}}} du

= log (u^{−1} (u + \sqrt{u^{2} + |y|^{2}})) |_{ϵ}^{(j+1)ϵ}

+ log (|u − ϵ| (u + \sqrt{u^{2} + |y|^{2}})) |_{−jϵ}^{0} + log (u + \sqrt{u^{2} + |y|^{2}}) |_{0}^{ϵ}.

Evaluating this and letting j → ∞, one obtains

\frac{1}{ϵ} lim_{j→∞} ∫_{0}^{ϵ} S_{j}du = \frac{1}{4πϵ}(log 2 + 2 log ϵ − log \frac{|y|^{2}}{2} )

from which we conclude that

\frac{1}{ϵ} ∫_{0}^{ϵ} V_{0}du = −\frac{1}{4πϵ} log |y|^{2}.

To compute the other terms in the Fourier expansion, we just need to calculate

\frac{1}{ϵ} ∫_{0}^{ϵ} V_{0}e^{2πimu/ϵ}du = \frac{1}{4πϵ} ∫_{−∞}^{∞} \frac{e^{2πimu/ϵ}}{\sqrt{u^{2} + |y|^{2}}} du

= \frac{1}{2πϵ} ∫_{0}^{∞} \frac{cos(2πmu/ϵ)}{\sqrt{u^{2} + |y|^{2}}}du

= \frac{1}{2πϵ} ∫_{0}^{∞} \frac{cos(2π|my|v/ϵ)}{\sqrt{v^{2} + 1}}dv

= \frac{1}{2πϵ}K_{0}(2π|my|/ϵ).

The last equality follows from [3], page 376, formula 9.6.21.

(c) By [3], 9.8.6, there exists a constant C_{1} such that \sqrt{x}e^{x}K_{0}(x) ≤ C_{1} for x ≥ 2. (In fact C_{1} ≤ 2). In particular, K_{0}(x) ≤ C_{1}e^{−x} for x ≥ 2. Thus

| ∑_{{m=−∞ \atop m≠0}}^{∞} \frac{1}{2πϵ}e^{2πimu/ϵ}K_{0}(2π|my|/ϵ)| ≤ \frac{C_{1}}{πϵ} ∑_{m=1}^{∞} e^{−2π|my|/ϵ}

= \frac{C_{1}}{πϵ} \frac{e^{−2π|y|/ϵ}}{1 − e^{−2π|y|/ϵ}}

for 2π|y|/ϵ ≥ 2. From this follows (c).

(d) By the maximum principal, the minimum value M of f occurs on the boundary of D_{r}. On the other hand, for fixed u, it is clear V_{0} is monotonically decreasing in |y|. Thus the minimum value of V_{0} + f/ϵ must occur on (∂D_{r}) × R. But taking r_{0} < r, by (c) there exists an ϵ_{0} such that for all ϵ < ϵ_{0},

|V_{0} + \frac{1}{4πϵ} log |y|^{2}| < −\frac{1}{4πϵ} log r^{2} + M/ϵ

whenever |y| = r. Thus V_{0} + f/ϵ is positive on ∂D_{r} × R for ϵ < ϵ_{0}, hence V_{0} + f/ϵ is positive on D_{r} × R. q.e.d.

With this rather technical lemma out of the way, we may now proceed to the construction of our metric, using the Gibbons–Hawking Ansatz formalism, as developed in §2. Suppose D_{r} ⊂ C is the disc of radius r < 1, centre the origin, and f : X̄ → D_{r} an elliptic fibration, with singular fibre over the origin of type I_{1}. Let Ȳ = D_{r} × R/ϵZ and Y = (D_{r} × R − {0} × ϵZ)/ϵZ. It is straightforward to check that there is an induced map π̄ : X̄ → Ȳ of C^{∞} manifolds, which restricts to an S^{1}-bundle π : X → Y with Chern class ±1, the sign dependent on the choice of orientation for the fibre. For further justification of these statements, the reader is referred to [15], Example 2.6 (1). The plan now is to define a hyperkähler metric on X via the Gibbons–Hawking Ansatz applied to π : X → Y, and then check that it extends to a hyperkähler metric on X̄.

Proposition 3.2. With the notation as above, let

h(y) = f(y_{1}, y_{2}) + ig(y_{1}, y_{2})

be a holomorphic function on D_{r}, so that

−\frac{1}{4π} log |y|^{2} + f(y_{1}, y_{2}) > 0

on D_{r}. Let V_{0} be the harmonic function on Y defined in Lemma 3.1, and V = V_{0} + f(y_{1}, y_{2})/ϵ, with ϵ chosen small enough so that V > 0 on Y. Then there exists a connection 1-form θ on X such that dθ/2πi = ∗dV, and this defines a hyperkähler metric on X with

− Re Ω = dy_{1} ∧ θ/2πi + V dy_{2} ∧ du
− Im Ω = dy_{2} ∧ θ/2πi + V du ∧ dy_{1}
ω = du ∧ θ/2πi + V dy_{1} ∧ dy_{2}.

These forms extend to X̄, giving a hyperkähler metric on X̄, and a holomorphic elliptic fibration X̄ → D_{r} with periods 1 and \frac{1}{2πi} log y + ih(y) + C, for some real constant C. By appropriate choice of θ, this constant C may be taken to be zero.

Proof. Since V is harmonic, recall that ∗dV is closed. Taking a sphere S^{2} of radius < ϵ centred at 0 ∈ D_{r} × R, we have

∫_{S^{2}} ∗dV = ∫_{S^{2}} ∗d (\frac{1}{4π\sqrt{u^{2} + y_{1}^{2} + y_{2}^{2}}} )

since all other terms in ∗dV are defined at 0, and hence are exact on an ϵ-ball around 0, and therefore do not contribute to the integral. In Example 2.5, it was however observed that this latter integral is ±1 (depending on the orientation of the sphere). Thus, since a connection form θ can be found such that idθ/2π is any desired representative of c_{1}, we can find a connection form θ such that dθ/2πi = ∗dV. Applying now the Gibbons–Hawking Ansatz construction described in §2, we obtain a hyperkähler metric on X, with the forms Re Ω, Im Ω and ω as described in the Proposition.

To see that these forms extend to X̄, focus on an ϵ/2-ball B around 0 in Ȳ. Then π̄^{−1}(B) → B can be identified with the map given in Example 2.5, restricted to the inverse image of the ϵ/2-ball in C^{2}. Let θ′ be the connection form given in that example. Now d(θ) − d(θ′) is the pull-back of an exact form on B, since all other terms of V besides the n = 0 term are defined on B. Thus on π̄^{−1}(B − {0}) we can write θ = θ′ + π̄^{∗}β for a form β defined on all of B. Now consider for example on π̄^{−1}(B − {0})

ω = du ∧ θ/2πi + V dy_{1} ∧ dy_{2}

= du ∧ (θ′ + π̄^{∗}β)/2πi + (1/4π\sqrt{u^{2} + y_{1}^{2} + y_{2}^{2}} + V′) dy_{1} ∧ dy_{2}

where V′ is a function defined everywhere on B. Thus we obtain

du ∧ \frac{θ′}{2πi} + (1/4π\sqrt{u^{2} + y_{1}^{2} + y_{2}^{2}}) dy_{1} ∧ dy_{2} + du ∧ \frac{β}{2πi} + V′dy_{1} ∧ dy_{2}.

The sum of the first two terms was seen to extend to all of π̄^{−1}(B) in Example 2.5, and the last two terms are defined everywhere on B, so ω extends to X̄. Note that ω^{2} ≠ 0 at the singular point of the singular fibre, because both du ∧ β and V′dy_{1} ∧ dy_{2} vanish at that point.

Finally, we compute the periods. Referring back to our discussion of S^{1}-invariant Ricci-flat metrics in §2, one of the periods is constant, value 1. The other period τ(y) is locally holomorphic in y, and given by

∫_{γ_{2}} dx = ∫_{γ_{2}} θ_{0} − i ∫_{γ_{2}} V du,

where γ_{2} is an S^{1} in the fibre X_{y} mapping isomorphically to {y} × S^{1} ⊂ Y. Calculating the imaginary part of this,

∫_{γ_{2}} dx_{2} = − ∫_{γ_{2}} V du = ±(\frac{1}{4π} log |y|^{2} − f(y_{1}, y_{2})),

using the Fourier expansion for V_{0} proved in Lemma 3.1 (b). We choose the orientation of γ_{2} to obtain the choice of sign to be minus. Then ∫_{γ_{2}} dx_{1} is necessarily locally a harmonic conjugate of −\frac{1}{4π} log |y|^{2} + f(y_{1}, y_{2}), and so the period of γ_{2} is

\frac{1}{2πi} log y + ih(y) + C

for some real constant C. Now θ_{0} may be modified by adding a term adu (a ∈ R) without changing the fact that dθ_{0} = ∗dV. If θ_{0} is changed in this way, we have

∫_{γ_{2}} dx_{1} = ∫_{γ_{2}} θ_{0} + adu
= aϵ + ∫_{γ_{2}} θ_{0}.

We can therefore choose a suitably to obtain C = 0, and hence the periods as claimed. q.e.d.

Remark 3.3. (1) There is still some remaining flexibility over choosing θ, as we can change θ by the pull-back of a closed form from Ȳ. This however need not worry us, since in order to perform the gluing in §4, we will in any case need to twist the Ooguri–Vafa metrics, the twist given as translation by an appropriate local section.

(2) Recall that in the holomorphic canonical coordinates x, y, the holomorphic 2-form Ω on X is just dx∧dy, and so the complex structure on X will be the one desired. This 2-form extends uniquely to give the correct complex structure on X̄.

Remark 3.4. A useful transformation. As ϵ → 0, the behaviour near the singular fibre of the Ooguri–Vafa metric is understood best by making a change of variables. The periods may be assumed to be 1, \frac{1}{2πi} log y+ih(y), as in Proposition 3.2, and we take V = V_{0}+f(y)/ϵ. We make the change of variables s = u/ϵ, v_{1} = y_{1}/ϵ, v_{2} = y_{2}/ϵ. Thus the disc of radius ϵ in the complex y-plane corresponds to the unit disc in the complex v-plane. If we now consider V_{0} as a function of these new variables, we observe that

ϵV_{0} = \frac{1}{4π} ∑_{n=−∞}^{∞} (\frac{1}{\sqrt{(s + n)^{2} + v_{1}^{2} + v_{2}^{2}}} − c_{|n|}) ,

where c_{n} = \frac{1}{n} (n > 0), and

c_{0} = 2(−γ + log(2ϵ)) = 2(−γ + log 2) + 2 log ϵ.

So, if Ṽ_{0} is the standard function V_{0} in variables s, v_{1}, v_{2} for ϵ = 1, we deduce that

ϵV = Ṽ_{0} − \frac{1}{2π} log ϵ + f.

Thus, if we start with an Ooguri–Vafa metric with fibres of volume ϵ over the disc of radius ϵ, make the change of variables described above, and then rescale the metric by ϵ^{−1}, we obtain the Ooguri–Vafa metric over the unit disc, with fibres of volume one, corresponding to the harmonic function Ṽ_{0} +f − \frac{1}{2π} log ϵ. Thus the periods of the corresponding elliptic fibration are seen to be 1, \frac{1}{2πi} log v+ \frac{1}{2πi} log ϵ+ih(y). This transformation lies behind the various estimates for diameters and curvature we derive below. We note here that in fact the formula given in [29] was for ϵV_{0}, rather than V_{0}, except that the constant a_{0} was not specified. The exact value for a_{0} greatly influences the behaviour of the metric as ϵ → 0, so this is quite important.

To understand the metric for |y| > ϵ, we can use the Fourier expansion for V_{0} from Lemma 3.1 (b), and use the same change of variables as above. Thus

ϵV_{0} = −\frac{1}{4π} log(v_{1}^{2} + v_{2}^{2}) − \frac{1}{2π} log ϵ + ∑_{{m=−∞ \atop m≠0}}^{m=∞} \frac{1}{2π} e^{2πims}K_{0}(2π|mv|)

where v = v_{1} + iv_{2}.

Estimates for diameter and curvature

We now consider a fixed elliptic fibration f : X′ → D′ over a disc D′ of radius a′ < 1, with singular fibre of Type I_{1} over the origin, and which we assume extends to an elliptic fibration over some larger disc. We assume that the periods are of the form 1, τ(y), where τ(y) = \frac{1}{2πi} log y + ih(y) as in Proposition 3.2. We then wish to study sequences of Ooguri–Vafa metrics yielding the correct holomorphic 2-form Ω, but with the volume ϵ of the fibres tending to zero — such metrics exist on X′ for small enough ϵ by Proposition 3.2. We first ask about the diameters of the fibres.

Proposition 3.5. There exists a positive constant C_{1} (independent of ϵ) such that, for metrics as above with fibre volume ϵ, the diameters of the fibres over D′ are bounded above by C_{1}(ϵ log ϵ^{−1})^{1/2}. Moreover, there exists a second constant C_{2} such that the diameter d(ϵ) of the singular fibre is at least C_{2}(ϵ log ϵ^{−1})^{1/2}.

Remark 3.6. In particular, it follows that d(ϵ) → 0 as ϵ → 0. If however we rescale the metric by ϵ^{−1} as in Remark 3.4 to obtain fibres of volume one, then the diameter of the singular fibre is of order (log ϵ^{−1})^{1/2}, and therefore becomes arbitrarily large as ϵ → 0. This then contrasts with the situation for a non-singular fibre, where for sufficiently small ϵ, the Ooguri–Vafa metric near this non-singular fibre is close to being semi-flat. Thus the diameter of the fibre in the rescaled metric remains bounded.

Proof. To calculate the diameter of a fibre, we recall from §2 the formula for ds^{2} in the Gibbons–Hawking Ansatz, namely

ds^{2} = V du · du + V^{−1}θ_{0}^{2}.

From this, it is clear that the diameter of a fibre is at least ∫_{0}^{ϵ/2} V^{1/2}du = \frac{1}{2} ∫_{0}^{ϵ} V^{1/2}du. Recall however that for all y ≠ 0, there exists a point on the fibre over y at which V = Im τ(y)/ϵ, where now Im τ(y) is bounded below by a positive constant for y ∈ D′. For some constant C therefore, we have on each fibre 0 ≠ y ∈ D′, a point at which V^{−1/2} ≤ Cϵ^{1/2}; by continuity, this is also true for the singular fibre. Thus each fibre over D′ contains an S^{1} in the S^{1}-bundle (where u is constant) of length at most Cϵ^{1/2}, and hence the diameter of the fibre is at most \frac{1}{2} ∫_{0}^{ϵ} V^{1/2}du+Cϵ^{1/2}. Since V = V_{0} + f(y_{1}, y_{2})/ϵ, and |f| is bounded on D′ by some constant A > 0, we have

∫_{0}^{ϵ} V_{0}^{1/2}du − A^{1/2}ϵ^{1/2} ≤ ∫_{0}^{ϵ} V^{1/2}du ≤ ∫_{0}^{ϵ} V_{0}^{1/2}du + A^{1/2}ϵ^{1/2}.

Since ∫_{0}^{ϵ} V_{0}^{1/2}du clearly takes its maximum when y = 0, we are reduced to estimating ∫_{0}^{ϵ} V_{0}^{1/2}du on the singular fibre only, and showing that it is of order (ϵ log ϵ^{−1})^{1/2}.

We now let V̄_{0} denote the restriction of V_{0} to the singular fibre, that is we take y = 0. Making the substitution s = u/ϵ as above, we observe that, for 0 < s < 1,

4πϵV̄_{0} = ∑_{n=1}^{∞} (\frac{1}{s + n} − \frac{1}{n}) + ∑_{n=1}^{∞} (\frac{1}{−s + n} − \frac{1}{n}) + \frac{1}{s} + 2γ − 2 log 2ϵ.

We now quote formula 6.3.16 from [3], for the fact that, for 0 < s < 1,

− ∑_{n=1}^{∞} (\frac{1}{s + n} − \frac{1}{n}) − γ = ψ(1 + s),

where ψ denotes the psi function. Thus, for 0 < s < 1,

4πϵV̄_{0} = −ψ(1 + s) − ψ(1 − s) + \frac{1}{s} − 2 log 2ϵ.

Using formula 6.3.15 from [3], we know that

− (ψ(1 + s) + ψ(1 − s)) = 2(1−s^{2})^{−1} +2γ −2+ ∑_{n=1}^{∞} 2(ζ(2n+1)−1)s^{2n},

where ζ denotes the usual zeta function. Hence, for 0 < s < 1,

4πϵV̄_{0} = 2(1 − s^{2})^{−1} + \frac{1}{s} − 2 log ϵ + G + 2g(s),

with G = −2 log 2 + 2γ − 2, and where

g(s) = ∑_{n=1}^{∞}(ζ(2n + 1) − 1)s^{2n}

has radius of covergence at least 2 (by inspection of the coefficients), and so defines a continuous (non-negative) function on [0, 1]. Now observe that ∫_{0}^{ϵ} V̄_{0}^{1/2}du = ϵ^{1/2} ∫_{0}^{1}(ϵV̄_{0})^{1/2}ds. The lower bound now follows immediately by ignoring the first two terms in the expression for 4πϵV̄_{0}. The upper bound follows by using the elementary fact that for α, β nonnegative real numbers, (α+β)^{1/2} ≤ α^{1/2} +β^{1/2}, along with the fact that the integrals ∫_{0}^{1} s^{−1/2}ds and ∫_{0}^{1}(1 − s^{2})^{−1/2}ds are finite. q.e.d.

Corollary 3.7. With notation as in Proposition 3.5, we suppose D ⊂ D′ is a disc centred on the origin of radius a ≤ a′ < 1, and let Diam(ϵ) denote the diameter of the total space of the elliptic fibration over D, under an Ooguri–Vafa metric on X′ with fibre volume ϵ. There exists a constant C_{3} (independent of both ϵ and a) such that, if ϵ ≤ a, then

Diam(ϵ) < C_{3}a^{1/2}ϵ^{−1/2}.

Proof. Consider the slice u = ϵ/2 of Y, and a radial curve γ from y = 0 to y = ae^{iθ} within this slice. There is a horizontal lift γ̃ of γ to X; recalling that

ds^{2} = V du · du + V^{−1}θ_{0}^{2},

we deduce that the length of γ̃ is just

∫_{γ} V (y, ϵ/2)^{1/2}|dy| = ∫_{0}^{a} V (re^{iθ}, ϵ/2)^{1/2}dr.

Since by Proposition 3.5, the diameters of the fibres are bounded above by

C_{1}ϵ^{1/2}(log ϵ^{−1})^{1/2} < C_{1}a^{1/2}ϵ^{−1/2},

if we can show that the latter integral is bounded above by Ca^{1/2}ϵ^{−1/2}, for some constant C independent of both ϵ and a, then the desired bound for Diam(ϵ) will follow (to go between any two fibres, we can always take the route via the central fibre).

We estimate the above integral in two parts, from 0 to ϵ, and from ϵ to a. We can estimate the first of these integrals most easily by performing the useful transformation described in Remark 3.4. Recall that

ϵV = Ṽ_{0} + f − \frac{1}{2π} log ϵ.

Now Ṽ_{0}(|v|, 1/2) is bounded above for 0 ≤ |v| ≤ 1 by Ṽ_{0}(0, 1/2), and so ϵV (re^{iθ}, ϵ/2) ≤ A′ − \frac{1}{2π} log ϵ for 0 ≤ r ≤ ϵ, where A′ is some positive constant. Thus

∫_{0}^{ϵ} V (re^{iθ}, ϵ/2)^{1/2}dr ≤ ϵ^{−1/2} ∫_{0}^{ϵ}(A′ − \frac{1}{2π} log ϵ)^{1/2}dr
≤ C′ϵ^{1/2}(log ϵ^{−1})^{1/2},

for some positive constant C′ independent of ϵ (and of course a). We therefore now need to demonstrate that

∫_{ϵ}^{a} V (re^{iθ}, ϵ/2)^{1/2}dr

has a bound of the desired type. To do this, we use the expression for V_{0} given in Lemma 3.1(b). From the proof of Lemma 3.1(c), we deduce that, for |y| ≥ ϵ/π, we have

2πϵV_{0} < − log |y| + 2C′_{1} \frac{e^{−2π|y|/ϵ}}{1 − e^{−2π|y|/ϵ}} .

In particular, since the second term is decreasing in the range, we have, for |y| ≥ ϵ, that

2πϵV_{0} < − log |y| + 2C′_{1} \frac{e^{−2π}}{1 − e^{−2π}} ,

and hence that

2πϵV < − log |y| + C′_{2},

for some constant C′_{2} independent of ϵ and a. Using the assumption that a ≤ a′ < 1, we have

∫_{ϵ}^{a} V (re^{iθ}, ϵ/2)^{1/2}dr < (2πϵ)^{−1/2} ∫_{ϵ}^{a}(C′_{2} − log r)^{1/2}dr
< ϵ^{−1/2}C′_{3} ∫_{ϵ}^{a}(log r^{−1})^{1/2}dr
< ϵ^{−1/2}C′_{3} ∫_{ϵ}^{a} r^{−1/2}dr
< 2C′_{3}ϵ^{−1/2}a^{1/2},

for an appropriate constant C′_{3}, depending on a′ but independent of ϵ and a. The result then follows immediately. q.e.d.

Proposition 3.8. With notation as in Proposition 3.5, let R(ϵ) denote the curvature tensor of the total space X′ of the elliptic fibration over D′, under an Ooguri–Vafa metric on X′ with fibre volume ϵ. Then there exists positive constants C_{4}, C′_{4} (independent of ϵ) such that, for all sufficiently small ϵ,

C′_{4}ϵ^{−1} log(ϵ^{−1})^{−2} < ∥R(ϵ)∥_{C^{0}} < C_{4}ϵ^{−1} log(ϵ^{−1}),

where ∥ . ∥_{C^{0}} denotes the usual C^{0}-norm on X′.

Proof. Recall first from Remark 2.4 that

∥R∥^{2} = 12V^{−6}|∇V |^{4} + V^{−4}Δ(|∇V |^{2}) − 6V^{−5}(∇V) · (∇(|∇V |^{2})).

We now perform our change of coordinates s = u/ϵ, v = y/ϵ. We recall that V = V_{0} + f(y_{1}, y_{2})/ϵ for some bounded harmonic function f defined over D′, and that

ϵV = Ṽ_{0} + f − \frac{1}{2π} log ϵ.

Also observe that ∇_{u,y_{1},y_{2}} = ϵ^{−1}∇_{s,v_{1},v_{2}}; from now on ∇ will denote ∇_{s,v_{1},v_{2}}, and Δ will denote Δ_{s,v_{1},v_{2}}. We set V_{1} = ϵV = Ṽ_{0} + f − \frac{1}{2π} log ϵ, considered as a function of s, v_{1}, v_{2}. Thus

ϵ^{2}∥R∥^{2} = 12V_{1}^{−6}|∇V_{1}|^{4} + V_{1}^{−4}Δ(|∇V_{1}|^{2}) − 6V_{1}^{−5}(∇V_{1}) · (∇(|∇V_{1}|^{2})).

We first prove the upper bound for ∥R(ϵ)∥_{C^{0}}, namely that

∥R(ϵ)∥ < C_{4}ϵ^{−1} log(ϵ^{−1})

at all points of X′. The easy part of this is to deal with the points in the range 1/2 ≤ |v| < a/ϵ (where a now denotes the radius of D′), corresponding to |y| ≥ ϵ/2 in the disc D′. Here we use the Fourier expansion for V_{1}, namely

V_{1} = −\frac{1}{2π} log |v| + f − \frac{1}{2π} log ϵ + ∑_{{m=−∞ \atop m≠0}}^{m=∞} \frac{1}{2π} e^{2πims}K_{0}(2π|mv|).

Recalling that K_{0}(x) and its derivatives decay at least as fast as e^{−x} for large x, it is clear that |∇V_{1}|^{4}, Δ(|∇V_{1}|^{2}) and (∇V_{1}) · (∇(|∇V_{1}|^{2})) are bounded (independent of ϵ) for 1/2 ≤ |v| < a/ϵ. Moreover, for ϵ sufficiently small,

ϵV = −\frac{1}{2π} log |y| + f(y) + ∑_{{m=−∞ \atop m≠0}}^{m=∞} \frac{1}{2π} e^{2πimu/ϵ}K_{0}(2π|my|/ϵ)

is bounded below, over D′, by some positive constant (independent of ϵ). Thus, V_{1} is bounded below on 1/2 ≤ |v| < a/ϵ, and hence ϵ∥R∥ is bounded above on the given range by some constant, again independent of ϵ.

The trickier argument is of course for the range 0 ≤ |v| ≤ 1/2, corresponding to 0 ≤ |y| ≤ ϵ/2. We assume that ϵ is small enough that 3ϵ/4 ≤ a. We make our usual change of variables, so that

V_{1} = ϵV = Ṽ_{0} + f + \frac{1}{2π} log(ϵ^{−1})

defines an Ooguri–Vafa metric over the disc |v| < 3/4, fibres of volume one, and periods {1, (2πi)^{−1} log v+(2πi)^{−1} log ϵ+ih}. Now choose A ≥ 0 such that f + A > 0 whenever |v| < 3/4, and set V_{2} = Ṽ_{0} + f + A; V_{2} then determines an Ooguri–Vafa metric over the disc |v| < 3/4, fibres of volume one, and periods {1, (2πi)^{−1} log v + ih + iA}. We may obviously assume that A < (2π)^{−1} log(ϵ^{−1}). Let R_{1}, respectively R_{2}, denote the curvature tensors of the metrics determined by V_{1}, respectively V_{2}. Our aim now is to show that ∥R_{1}∥^{2} < C log(ϵ^{−1})^{2} over the disc |v| ≤ 1/2; if this is true, it follows from the above that ∥R∥ < C_{4}ϵ^{−1} log(ϵ^{−1}) at all points over D′, for some positive constant C_{4}.

Since the metric determined by V_{2} is independent of ϵ, it is clear that ∥R_{2}∥ is bounded over |v| ≤ 1/2, with the bound independent of ϵ. Hence

(3.1) 12V_{2}^{−6}|∇V_{2}|^{4} + V_{2}^{−4}Δ(|∇V_{2}|^{2}) − 6V_{2}^{−5}(∇V_{2}) · (∇(|∇V_{2}|^{2}))

is bounded independent of ϵ over the range in question, |v| ≤ 1/2, which from now on will be taken as understood. We wish to show that

∥R_{1}∥^{2} = 12V_{1}^{−6}|∇V_{2}|^{4} + V_{1}^{−4}Δ(|∇V_{2}|^{2})
− 6V_{1}^{−5}(∇V_{2}) · (∇(|∇V_{2}|^{2}))
≤ C(log(ϵ^{−1}))^{2}.

Since V_{1} = V_{2} + (2π)^{−1} log(ϵ^{−1}) − A ≥ V_{2}, it will be enough to prove the same bound for

\begin{matrix}12V_{1}^{−2}V_{2}^{−4}|∇V_{2}|^{4} + V_{2}^{−4}Δ(|∇V_{2}|^{2}) \\ − 6V_{1}^{−1}V_{2}^{−4}(∇V_{2}) · (∇(|∇V_{2}|^{2})). \end{matrix} (3.2)

By subtracting our previously bounded expression (3.1), we need then only show boundedness for

\begin{matrix}12V_{1}^{−2}V_{2}^{−6}(V_{2}^{2} − V_{1}^{2})|∇V_{2}|^{4} \\ − 6V_{1}^{−1}V_{2}^{−5}(V_{2} − V_{1})(∇V_{2}) · (∇(|∇V_{2}|^{2})). \end{matrix} (3.3)

Expanding this latter expression out, we get

\begin{aligned}6 ((2π)^{−1} log(ϵ^{−1}) − A) V_{1}^{−1} & (V_{2}^{−5}(∇V_{2}) · (∇(|∇V_{2}|^{2})). \\ & − 2V_{2}^{−6}(1 + \frac{V_{2}}{V_{1}} )|∇V_{2}|^{4}). \end{aligned} (3.4)

We now claim that

|V_{1}^{−1} (V_{2}^{−5}(∇V_{2}) · (∇(|∇V_{2}|^{2})) − 4V_{2}^{−6}|∇V_{2}|^{4})|

and

V_{1}^{−2}V_{2}^{−6}|∇V_{2}|^{4}

are bounded independent of ϵ. If this is true, then the latter bound will imply that

V_{1}^{−1}(1 − \frac{V_{2}}{V_{1}} )V_{2}^{−6}|∇V_{2}|^{4} ≤ C′(log(ϵ^{−1}) − 2πA)

for some positive C′, and then the former bound implies that the expression we are interested in has a bound of the form

B_{1}(log(ϵ^{−1}) − 2πA)^{2} + B_{2}(log(ϵ^{−1}) − 2πA),

for suitable positive constants B_{1}, B_{2}. This then gives the required result.

To show boundedness for the two remaining quantities, it is sufficient to bound the functions

V_{2}^{−6} |4V_{2}^{−1}|∇V_{2}|^{4} − (∇V_{2}) · (∇(|∇V_{2}|^{2}))|

and

V_{2}^{−8}|∇V_{2}|^{4}.

Both these functions are defined away from {0} × Z and are periodic in s; moreover, they plainly do not depend on ϵ. If we show that they are in fact both continuous at the origin (v = 0, s = 0), the existence of the required bounds will follow automatically.

We now write 4πV_{2} = ρ^{−1} + w, where ρ = (s^{2} + v_{1}^{2} + v_{2}^{2})^{1/2} and w is a harmonic function on a neighbourhood of the origin. Then we see that (4π)^{4}|∇V_{2}|^{4} = ρ^{−8} + O(ρ^{−6}). Since (4πV_{2})^{−8} = ρ^{8}(1 + wρ)^{−8}, we deduce that V_{2}^{−8}|∇V_{2}|^{4} is regular at the origin, taking the value (4π)^{4} there. Moreover, it is easily checked that

(4π)^{3}(∇V_{2}) · (∇(|∇V_{2}|^{2})) = 4ρ^{−7} + O(ρ^{−5}),

and so in particular

4V_{2}^{−7}|∇V_{2}|^{4} − V_{2}^{−6}(∇V_{2}) · (∇(|∇V_{2}|^{2}))

is also regular at the origin, and vanishes there.

We now turn to the lower bound for ∥R(ϵ)∥_{C^{0}}. We work on the transformed elliptic fibration over the disc |v| ≤ 1/2, and let M denote the C^{0}-norm of the function given by (3.1).

From the above calculations, at all points P with sufficiently small value of ρ, we have

|V_{2}^{−6}(∇V_{2}) · (∇(|∇V_{2}|^{2}) − 4V_{2}^{−7}|∇V_{2}|^{4}| < M , V_{2}^{−7}|∇V_{2}|^{4} > 2M.

We now fix such a point P; note that the coordinates s, v_{1}, v_{2} are then taken to be fixed, and so this does not correspond to taking a fixed point (independent of ϵ) on our original family X′.

Observe now that

V_{1}/V_{2} = 1 + \frac{(2π)^{−1} log(ϵ^{−1}) − A}{V_{2}};

so for P fixed, V_{1}(P)/V_{2}(P) > 2 for ϵ sufficiently small. From this it follows that, when evaluated at P,

|V_{2}^{−6}(∇V_{2}) · (∇(|∇V_{2}|^{2})) − 2V_{2}^{−7}(1 + \frac{V_{2}}{V_{1}} )|∇V_{2}|^{4}| > M,

for ϵ sufficiently small. Hence, for ϵ sufficiently small, the modulus of (3.4) evaluated at P is at least 3M say, and thus the same is true of (3.3). From this, and our original choice for M, it follows that the modulus of (3.2) evaluated at P is at least 2M. Therefore, for ϵ sufficiently small,

∥R_{1}(P)∥^{2} > B(log(ϵ^{−1}))^{−4}

for some constant B independent of ϵ. Thus

∥R(ϵ)∥_{C^{0}} > B^{1/2}ϵ^{−1}(log(ϵ^{−1}))^{−2},

as required. q.e.d.

4. Almost Ricci-flat metrics on elliptic K3 surfaces

Our goal in this section is to construct Kähler metrics on elliptic K3 surfaces which are very close to being Ricci-flat by gluing the Ooguri– Vafa metric in neighbourhoods of singular fibres to the semi-flat metric away from the singular fibres.

We begin by producing one such metric on a Jacobian elliptic fibration. Fix a K3 surface X with a fixed holomorphic 2-form Ω and an elliptic fibration f : X → B = P^{1}, which we will take to have a holomorphic section σ_{0}. Furthermore, assume all singular fibres of f are of Kodaira type I_{1}; there will then be 24 such fibres. Let p_{1}, …, p_{24} ∈ B be those points for which X_{p_{i}} = f^{−1}(p_{i}) is singular, Δ = {p_{1}, …, p_{24}}, B_{0} = B − Δ, X_{0} = f^{−1}(B_{0}), and X^{#} = X − Sing(f^{−1}(Δ)). There is an exact sequence, already mentioned in §1,

0⟶R^{1}f_{∗}Z⟶T_{B}^{∗}⟶^{ϕ}X^{#}⟶0,

with the property that ϕ maps the zero section of T_{B}^{∗} to σ_{0} and ϕ^{∗}Ω is the canonical holomorphic 2-form on T_{B}^{∗}, which is dx∧dy if y is a coordinate on B and x a canonical fibre coordinate. (See [14], Proposition 7.2). Here x = 0 defines the zero section.

Given this data, by Example 2.2, for each ϵ, there exists a welldefined Ricci-flat metric on X_{0}, the standard semi-flat metric ω_{SF}, with the area of each fibre being ϵ. The reader should keep in mind the dependence of ω_{SF} on ϵ.

Now let y be a holomorphic coordinate on B defined in a neighbourhood U of p ∈ Δ, U contractible with U ∩ Δ = {p}, and y = 0 at the point p. Let x be the corresponding canonical fibre coordinate. Let U^{∗} = U − {p}, X_{U^{∗}} = f^{−1}(U^{∗}). We can then choose over U^{∗} holomorphic periods τ_{1}(y), τ_{2}(y), representing possibly multi-valued holomorphic sections of T_{U^{∗}}^{∗} generating the period lattice. Because the

monodromy about an I_{1} fibre in a suitable basis is (\begin{matrix}1 & 1 \\ 0 & 1 \end{matrix}), we can

take one of these, say τ_{1}, to be single-valued, though τ_{2} will be multivalued around the I_{1} fibre. We will always choose τ_{1} and τ_{2} so that Im(τ̄_{1}τ_{2}) > 0. Set

W_{0}(y) = 1/Im(τ̄_{1}τ_{2})
b_{0}(x, y) = −\frac{Im(τ_{2}x̄)∂_{y}τ_{1} + Im(τ̄_{1}x)∂_{y}τ_{2}}{Im(τ̄_{1}τ_{2})}

and

∂_{v} = W_{0}^{−1}∂_{x}
∂_{h} = ∂_{y} − b_{0}∂_{x}
ϑ_{v} = W_{0}(dx + b_{0}dy)
ϑ_{h} = dy

as in §2. The latter two 1-forms are well-defined on X_{U}^{∗}, so form a basis for (1, 0) forms. We denote by ¯∂_{v} et cetera the complex conjugates of the above as usual.

Lemma 4.1. Let ω be a real closed (1, 1) form on X_{U^{∗}}, with

ω = \frac{i}{2}(α ϑ_{v} ∧ ϑ̄_{v} + β ϑ_{h} ∧ ϑ̄_{v} + β̄ ϑ_{v} ∧ ϑ̄_{h} + γ ϑ_{h} ∧ ϑ̄_{h}).

There exists a function φ on X_{U^{∗}} such that ω = i∂ ¯∂φ if and only if ω represents the zero cohomology class on X_{U^{∗}} and

∫_{X_{b}} β dx_{1} ∧ dx_{2} = 0

for all b ∈ U^{∗}. Furthermore, for 0 < r_{1} < r_{2}, let U_{r_{1},r_{2}} = {y ∈ U | r_{1} < |y| < r_{2}\}. If r_{1} < r′_{1} < r′_{2} < r_{2} and \overline{U}_{r_{1},r_{2}} ⊆ U^{∗}, then there exists a constant C depending only on r_{1}, r_{2}, r′_{1}, r′_{2} and the periods of f over U_{r_{1},r_{2}} such that φ can be chosen with

∥φ∥′_{C^{k+2,α}} ≤ C(∥α∥_{C^{k,α}} + ∥β∥_{C^{k,α}} + ∥γ∥_{C^{k,α}}).

Here, we compute the C^{k,α} norm of a function on f^{−1}(U_{r_{1},r_{2}}) by thinking of them as functions on T_{U_{r_{1},r_{2}}}^{∗}, which we embed in C^{2} by the coordinates x and y. We can then use the standard C^{k,α} norms on a bounded open set of T_{U_{r_{1},r_{2}}}^{∗} which contains a fundamental domain of each fibre. The norm ∥ · ∥′_{C^{k,α}} denotes the similar norm of a function over U_{r′_{1},r′_{2}}.

Remark 4.2. We note that the definition of the C^{k,α} norm given above depends on the choice of holomorphic coordinate y and the bounded open set, but any two such norms will be equivalent.

Proof. Before beginning the proof, we observe from (2.4) and (2.3) that

∂ϑ̄_{v} =((∂_{v}W_{0})ϑ_{v} + (∂_{h}W_{0})ϑ_{h}) ∧ W_{0}^{−1}ϑ̄_{v}
+ W_{0}((∂_{v}b̄_{0})ϑ_{v} + (∂_{h}b̄_{0})ϑ_{h}) ∧ ϑ̄_{h}
¯∂ϑ̄_{v} =0.

(4.1)

Furthermore, locally for the base, a function on X_{U}^{∗} can be expanded in a Fourier series on the fibres, yielding

f(x, y) = ∑_{n,m∈Z} a_{n,m}(y)e^{2πi(nIm(τ_{2}x̄)+mIm(τ̄_{1}x))/Im(τ̄_{1}τ_{2})}.

A direct calculation shows that

∂_{h}f = ∑_{n,m∈Z} ∂_{y}(a_{n,m})e^{2πi(nIm(τ_{2}x̄)+mIm(τ̄_{1}x))/Im(τ̄_{1}τ_{2})}.

If a φ exists, then of course ω represents the zero cohomology class on X_{U^{∗}}. Also,

i∂ ¯∂φ = i∂((¯∂_{v}φ)ϑ̄_{v} + (¯∂_{h}φ)ϑ̄_{h}).

From (4.1) it then follows that if ω = i∂ ¯∂φ, then

β = 2(∂_{h} ¯∂_{v}φ + (¯∂_{v}φ)W_{0}^{−1}∂_{h}W_{0}),

and then by looking at the constant term a_{0,0} of the Fourier expansion of β, it is clear that ∫_{X_{b}} βdx_{1} ∧ dx_{2} = 0.

Conversely, first suppose ω is cohomologically trivial. Then there exists a one-form ξ of type (1, 0) such that \frac{i}{2}d(ξ̄ − ξ) = ω (since ω is real). Necessarily ∂ξ = ¯∂ξ̄ = 0. Thus ξ̄ represents a class in H^{0,1}(X_{U^{∗}}). If this class is zero, then there exists a function φ such that ¯∂φ = ξ̄, and then ∂φ̄ = ξ, so

ω = \frac{i}{2}(∂ ¯∂φ − ¯∂∂φ̄) = \frac{i}{2}∂ ¯∂(φ + φ̄) = i∂ ¯∂Re φ,

as desired. Thus we need to understand when ξ̄ represents the zero class.

Now H^{0,1}(X_{U^{∗}}) = H^{1}(X_{U^{∗}}, O_{X_{U^{∗}}}), which by the Leray spectral sequence for f is isomorphic to H^{0}(U^{∗}, R^{1}f_{∗}O_{X_{U^{∗}}}), as H^{i}(U^{∗}, f_{∗}O_{X_{U^{∗}}}) = H^{i}(U^{∗}, O_{U^{∗}}) = 0 for i ≥ 1. Thus ξ̄ represents zero in H^{0,1}(X_{U^{∗}}) if and only if ξ̄|_{X_{b}} represents the zero class in H^{0,1}(X_{b}) for all b ∈ U^{∗}. If we write ξ̄ = ḡϑ̄_{v} + h̄ϑ̄_{h}, this is equivalent to the constant term in the Fourier expansion of ḡ on the fibre being zero. Denote this constant term by ḡ_{0}(y).

What kind of function is ḡ_{0}? Well, by (4.1),

0 = ¯∂ξ̄ = (¯∂_{h}ḡ − ¯∂_{v}h̄)ϑ̄_{h} ∧ ϑ̄_{v}.

By looking at the constant term of the Fourier expansion of this coefficient, we see ∂ȳḡ_{0} = 0, so ḡ_{0} is a holomorphic function on U^{∗}. This function gives the section of R^{1}f_{∗}O_{X_{U^{∗}}} defined by ξ̄.

Now let us compute the coefficient β of ϑ_{h} ∧ ϑ̄_{v} in ω in terms of g and h. From ω = \frac{i}{2}(∂ξ̄− ¯∂ξ) and (4.1), it follows that

β = ∂_{h}ḡ + ¯∂_{v}h + ḡW_{0}^{−1}∂_{h}W_{0} + gW_{0} ¯∂_{v}b_{0}.

If β_{0} is the constant term in the Fourier expansion of β, then we get, using (2.2′) for the second line,

\begin{aligned}β_{0} & = ∂_{y}ḡ_{0} + ḡ_{0}W_{0}^{−1}∂_{h}W_{0} + g_{0}W_{0} ¯∂_{v}b_{0} \\ & = ∂_{y}ḡ_{0} + ḡ_{0}∂_{x}b_{0} + g_{0}∂x̄b_{0} \\ & = ∂_{y}ḡ_{0} + \frac{ḡ_{0}(τ̄_{2}∂_{y}τ_{1} − τ̄_{1}∂_{y}τ_{2}) − g_{0}(τ_{2}∂_{y}τ_{1} − τ_{1}∂_{y}τ_{2})}{τ̄_{1}τ_{2} − τ_{1}τ̄_{2}} \\ & = ∂_{y}ḡ_{0} + b_{0}(ḡ_{0}, y). \end{aligned} (4.2)

If we now assume in addition that ∫_{X_{b}} βdx_{1} ∧ dx_{2} = 0 for all b ∈ U^{∗}, then β_{0} = 0, so

∂_{y}ḡ_{0} + b_{0}(ḡ_{0}, y) = 0.

Now write ḡ_{0}(y) = a_{1}(y)τ_{1}(y) + a_{2}(y)τ_{2}(y), where a_{1}, a_{2} are real functions of y. Then b_{0}(ḡ_{0}, y) = −a_{1}∂_{y}τ_{1} − a_{2}∂_{y}τ_{2}, so

0 = ∂_{y}ḡ_{0} + b_{0}(ḡ_{0}, y) = (∂_{y}a_{1})τ_{1} + (∂_{y}a_{2})τ_{2}.

But

0 = ∂ȳḡ_{0} = (∂ȳa_{1})τ_{1} + (∂ȳa_{2})τ_{2}.

Thus combining these two equations gives

(∂_{y_{1}}a_{1})τ_{1} + (∂_{y_{1}}a_{2})τ_{2} = 0
(∂_{y_{2}}a_{1})τ_{1} + (∂_{y_{2}}a_{2})τ_{2} = 0,

and by linear independence of τ_{1} and τ_{2} we see a_{1} and a_{2} are constant. Since ḡ_{0} is well-defined and we are assuming τ_{1} is the monodromy invariant period, we have ḡ_{0} = aτ_{1}, a a constant. Now a calculation shows that

\frac{i}{2}d(τ_{1}ϑ̄_{v} − τ̄_{1}ϑ_{v}) = 0,

so we can subtract aτ_{1}ϑ̄_{v} from ξ̄ without affecting \frac{i}{2}d(ξ̄− ξ) = ω. Thus we can assume ḡ_{0} = 0, and then ξ̄represents the zero class in H^{0,1}(X_{U^{∗}}), allowing us to complete the proof of the existence of φ.

Now we need to control the norm of φ. First note that

W_{0}^{2}α = 2∂_{x}∂x̄φ = \frac{1}{2}Δ_{x}φ,

where Δ_{x} = ∂_{x_{1}}^{2} + ∂_{x_{2}}^{2} denotes the standard Laplacian on fibres. Writing φ = φ_{0} + φ_{v} where φ_{0} is the pull-back of a function on U^{∗} and ∫_{X_{b}} φ_{v}dx_{1}dx_{2} = 0 for all b ∈ U^{∗}, we have W_{0}^{2}α = Δ_{x}φ_{v}/2. It then follows that |φ_{v}| is bounded on each fibre (being a torus) with the bound proportional to a bound for |α| on that fibre, with the constant of proportionality depending on the periods at that point. (To see this, one can just work with Fourier series). Thus

∥φ_{v}∥_{C^{0}} ≤ C_{1}∥α∥_{C^{0}}

on U_{r_{1},r_{2}}, where C_{1} depends on the periods over U_{r_{1},r_{2}}.

Next restrict φ and ω to the zero section of f : X_{U} → U. On this zero-section,

ω = \frac{i}{2}γ dy ∧ dȳ

= i∂ ¯∂φ

= i∂_{y}∂ȳφ dy ∧ dȳ

= \frac{i}{4}Δ_{y}φ dy ∧ dȳ.

so Δ_{y}φ = 2γ on the zero section, where Δ_{y} is the the standard Laplacian ∂_{y_{1}}^{2} + ∂_{y_{2}}^{2} on U^{∗}.

Now let ψ be a harmonic function on U_{r_{1},r_{2}} such that ψ|_{∂\overline{U}_{r_{1},r_{2}}} = φ|_{∂\overline{U}_{r_{1},r_{2}}} . (Here we are identifying U_{r_{1},r_{2}} with its image under the zero section.) This function exists and is unique. Then

Δ_{y} (φ − ψ + \frac{∥2γ∥_{C^{0}}}{4} (y_{1}^{2} + y_{2}^{2})) = 2γ + ∥2γ∥_{C^{0}} ≥ 0.

Thus by the maximum principal, φ − ψ + ∥2γ∥_{C^{0}}(y_{1}^{2} + y_{2}^{2})/4 achieves its maximum when either |y| = r_{1} or |y| = r_{2}, and since φ − ψ = 0 on the boundary of U_{r_{1},r_{2}}, we have

φ − ψ ≤ ∥2γ∥_{C^{0}} r_{2}^{2}/4.

Similarly, ψ − φ ≤ ∥2γ∥_{C^{0}} r_{2}^{2}/4, so

∥φ − ψ∥_{C^{0}} ≤ ∥γ∥_{C^{0}} r_{2}^{2}.

This estimate holds on U_{r_{1},r_{2}}, but from ∥φ_{v}∥_{C^{0}} ≤ C_{1}∥α∥_{C^{0}}, it is clear that the oscillation of φ along the fibres is bounded by C_{1}∥α∥_{C^{0}}, and thus on f^{−1}(U_{r_{1},r_{2}}),

∥φ − ψ∥_{C^{0}} ≤ C_{2}(∥α∥_{C^{0}} + ∥γ∥_{C^{0}})

for some constant C_{2} depending on the periods, r_{1}, and r_{2}. Noting that ∂ ¯∂ψ = 0, we can replace φ by φ − ψ. Then the C^{k+2,α} estimates follow from the standard interior Schauder estimates for the Laplacian (see [11], problem 6.1.) This is because the ordinary Laplacian (in the coordinates x_{1}, x_{2}, y_{1}, y_{2}) of φ can be expressed in terms of α, β and γ.

q.e.d.

Lemma 4.3. Let ω be a Kähler form on X_{U}, ω_{SF} the semi-flat Kähler form on X_{0}, such that

∫_{X_{b}} ω = ∫_{X_{b}} ω_{SF} = ϵ.

Then [ω_{SF} − ω] = 0 in H^{2}(X_{U^{∗}}, R), and furthermore, there exists a holomorphic section σ of f : X_{U} → U and a function φ on X_{U^{∗}} such that

ω_{SF} − T_{σ}^{∗}ω = i∂ ¯∂φ,

where T_{σ} is translation by the section σ.

Proof. To show the first part, we first observe that H_{2}(X_{U^{∗}}, Z) is generated by the homology classes of two submanifolds: X_{b} for some b ∈ U^{∗}, and T, where T is a torus fibred in circles over a simple closed loop γ : [0, 1] → U^{∗} generating π_{1}(U^{∗}), with the class of the fibre being the monodromy invariant cycle. To show [ω_{SF} − ω] = 0, we just need ∫_{X_{b}} ω_{SF} − ω = 0, which is obvious, and ∫_{T} ω_{SF} − ω = 0. Now ∫_{T} ω = 0 since ω is defined on X_{U}, where T is homologous to zero. On the other hand, if we describe T explicitly, parametrised by coordinates s, t with μ : [0, 1]^{2} → X_{U^{∗}} given by

μ(s, t) = (x(s, t), y(s, t)) = (sτ_{1}(γ(t)), γ(t)),

then a calculation shows that μ^{∗}ω_{SF} = 0, and hence ∫_{T} ω_{SF} = 0. Thus [ω_{SF} − ω] = 0.

As in Lemma 4.1, write, for each section σ of f : X_{U^{∗}} → U^{∗},

ω_{SF} − T_{σ}^{∗}ω = \frac{i}{2}(α_{σ}ϑ_{v} ∧ ϑ̄_{v} + β_{σ}ϑ_{h} ∧ ϑ̄_{v} + ⋯).

Let σ_{0} be the zero section, so that T_{σ_{0}} is the identity. We showed in (4.2) that the function β_{0}, the constant term in the Fourier expansion of β_{σ_{0}}, was of the form

β_{0} = ∂_{y}k + b_{0}(k, y)

where k(y) is a holomorphic function on U^{∗}.

Now write

ω = \frac{i}{2}(W W_{0}^{−2}ϑ_{v} ∧ ϑ̄_{v} + β_{ω}ϑ_{h} ∧ ϑ̄_{v} + ⋯)

where necessarily the constant term of W is

(Imτ̄_{1}τ_{2})^{−1} ∫_{X_{b}} W dx_{1} ∧ dx_{2} = (Imτ̄_{1}τ_{2})^{−1} ∫_{X_{b}} \frac{i}{2}W W_{0}^{−2}ϑ_{v} ∧ ϑ̄_{v}
= ϵ/Im(τ̄_{1}τ_{2}).

We calculate T_{σ}^{∗}ω. First note that

T_{σ}^{∗}(ϑ_{v}) = W_{0}(d(x + σ(y)) + b_{0}(x + σ(y), y) dy)
= W_{0}(dx + b_{0} dy) + W_{0}(∂_{y}σ(y) + b_{0}(σ(y), y)) dy
= ϑ_{v} + W_{0}(∂_{y}σ + b_{0}(σ(y), y)) ϑ_{h}.

Thus the coefficient of ϑ_{h} ∧ ϑ̄_{v} in T_{σ}^{∗}(ω) is

\frac{i}{2} (β_{ω} ∘ T_{σ} + \frac{W}{W_{0}} (∂_{y}σ + b_{0}(σ(y), y))) .

On the other hand, ω_{SF} = \frac{i}{2}W_{0}^{−1}(ϵ ϑ_{v} ∧ ϑ̄_{v} + ϵ^{−1} ϑ_{h} ∧ ϑ̄_{h}). Thus β_{σ_{0}} = −β_{ω}, and

β_{σ} = β_{σ_{0}} ∘ T_{σ} − \frac{W}{W_{0}} (∂_{y}σ + b_{0}(σ(y), y)).

So

(Imτ̄_{1}τ_{2})^{−1} ∫_{X_{b}} β_{σ} dx_{1} ∧ dx_{2}

= (Imτ̄_{1}τ_{2})^{−1}( ∫_{X_{b}} β_{σ_{0}} ∘ T_{σ} dx_{1} ∧ dx_{2}

− W_{0}^{−1}(∂_{y}σ + b_{0}(σ(y), y)) ∫_{X_{b}} W dx_{1} ∧ dx_{2})

= β_{0} − ϵ(∂_{y}σ + b_{0}(σ(y), y)).

If we take σ(y) = k(y)/ϵ, this will yield zero. So for this choice of σ, ω_{SF} − T_{σ}^{∗}ω = i∂ ¯∂φ for some function φ on X_{U^{∗}}.

Note that a holomorphic section of f over U^{∗} always extends to a holomorphic section of f on U. q.e.d.

Theorem 4.4. Let f : X → P^{1} be an elliptically fibred K3 surface with a holomorphic section and 24 singular fibres over Δ = {p_{1}, …, p_{24}} as above. Then there exists open sets U_{1}^{i} ⊆ U_{2}^{i} ⊆ P^{1}, i = 1, …, 24, each diffeomorphic to a disc, U_{j}^{i} ∩ Δ = {p_{i}}, positive constants D_{1}, …, D_{6} and ϵ_{0} such that, for all ϵ < ϵ_{0}, there exists a Kähler metric ω_{ϵ} on X with the following properties:

(1) ∫_{X} ω_{ϵ}^{2} = ∫_{X}(Re Ω)^{2} = ∫_{X}(Im Ω)^{2}.

(2) ∫_{X_{b}} ω_{ϵ} = ϵ.

(3) ω_{ϵ}|_{f^{−1}(P^{1}\⋃_{i} U_{2}^{i})} = ω_{SF}.

(4) ω_{ϵ}|_{f^{−1}(U_{1}^{i})} = T_{σ_{i}}^{∗}ω_{OV} , where ω_{OV} is an Ooguri–Vafa metric and T_{σ_{i}} denotes translation by some holomorphic section σ_{i}.

(5) If F_{ϵ} = log (\frac{Ω∧Ω̄/2}{ω_{ϵ}^{2}}), then

∥F_{ϵ}∥_{C^{0}} ≤ D_{1}e^{−D_{2}/ϵ}

and

∥ΔF_{ϵ}∥_{C^{0}} ≤ D_{1}e^{−D_{2}/ϵ}

where Δ denotes the Laplacian with respect to the metric ω_{ϵ}.

(6) inf_{v}{Ric(v, v) | |v|_{ω_{ϵ}} = 1} ≥ −D_{3}e^{−D_{4}/ϵ}.

(7) With the Riemannian metric induced by ω_{ϵ}, Diam(X) ≤ D_{5}ϵ^{−1/2}.

(8) If R denotes the Riemann curvature tensor, then

∥R∥_{C^{0}} ≤ D_{6}ϵ^{−1} log ϵ^{−1},
∥R∥_{C^{0}} → ∞ as ϵ → 0,

and on any non-singular fibre, there exists a constant C depending on the fibre such that

∥R∥ ≤ Cϵ.

Proof. Let p ∈ Δ; we fix our attention near this point. Choosing a holomorphic coordinate y in a neighbourhood of p, we can express the holomorphic periods of f as τ_{1}(y), τ_{2}(y), where τ_{1} is taken to be single valued. In T_{B}^{∗}, this coincides with the holomorphic differential τ_{1}(y)dy. Locally, there exists a function g(y) with dg = τ_{1}(y)dy; since τ_{1}(p) ≠ 0, we can use g as a local holomorphic coordinate in a neighbourhood of p. Replacing y by g, we can then assume that τ_{1}(y) = 1 and also that y = 0 at p. By results of §3, we can then construct for all ϵ less than some ϵ_{0}, a metric ω_{OV} on f^{−1}(U), for some U = {y | |y| < r}, for some r which only depends on the period τ_{2} and ϵ_{0}, but not ϵ. Fix r_{1} < r_{2} < r, and let U_{i} = {y | |y| < r_{i}}. If p = p_{j}, we set U_{i}^{j} = U_{i}.

Remaining focused near p, let ψ : (0, (r_{2} + δ)^{2}) → [0, 1] be a fixed C^{∞} cut-off function, with ψ(r^{2}) = 1 for r ≤ r_{1}, ψ(r^{2}) = 0 for r ≥ r_{2}.

Now apply Lemma 4.3 with ω = ω_{OV}. Then there exists a holomorphic section σ of f over U, such that

ω_{SF} − T_{σ}^{∗}ω_{OV} = i∂ ¯∂φ

for some function φ on X_{U^{∗}}. We can then glue T_{σ}^{∗}ω_{OV} and ω_{SF} by

ω_{new} = ω_{SF} − i∂ ¯∂(ψ(|y|^{2})φ).

For |y| ≥ r_{2}, ω_{new} coincides with ω_{SF}; for |y| ≤ r_{1}, ω_{new} coincides with T_{σ}^{∗}ω_{OV}. This can be done at each singular fibre, obtaining a global closed real (1, 1) form ω_{new}.

We still need to check ω_{new} is positive. One calculates that on X_{U^{∗}},

ω_{new} =(1 − ψ(|y|^{2}))ω_{SF} + ψ(|y|^{2})T_{σ}^{∗}ω_{OV} − i(ψ′(|y|^{2})ȳ dy ∧ ¯∂φ
+ ψ′(|y|^{2})y ∂φ ∧ dȳ + ψ′′(|y|^{2})|y|^{2}φ dy ∧ dȳ).

The sum of the first two terms is positive, so we need to make sure the last three terms are small. Thus we need to control the size of φ. To do so, we need to show ω_{SF} − T_{σ}^{∗}ω_{OV} is small. Now

ω_{SF} = \frac{i}{2}W_{0}^{−1}(ϵ ϑ_{v} ∧ ϑ̄_{v} + ϵ^{−1} ϑ_{h} ∧ ϑ̄_{h}).

On the other hand, we can assume σ is the zero section by having chosen the right holomorphic section in Construction 2.6 to perform the transformation between coordinates, and write, with W = V^{−1},

ω_{OV} =\frac{i}{2}(W (dx + b dy) ∧ \overline{(dx + b dy)} + W^{−1} dy ∧ dȳ)
=\frac{i}{2}(W (dx + b_{0} dy) ∧ \overline{(dx + b_{0} dy)} + W (b − b_{0}) dy ∧ \overline{(dx + b_{0} dy)}

+ W (b̄ − b̄_{0})(dx + b_{0} dy) ∧ dȳ + (W |b − b_{0}|^{2} + W^{−1}) dy ∧ dȳ)

=\frac{i}{2}(W W_{0}^{−2} ϑ_{v} ∧ ϑ̄_{v} + \frac{W}{W_{0}} (b − b_{0}) ϑ_{h} ∧ ϑ̄_{v}

+ \frac{W}{W_{0}} (b̄ − b̄_{0})(ϑ_{v} ∧ ϑ̄_{h}) + (W |b − b_{0}|^{2} + W^{−1}) ϑ_{h} ∧ ϑ̄_{h}).

Thus we are applying Lemma 4.1 with α = ϵW_{0}^{−1} − W W_{0}^{−2}, β = \frac{W}{W_{0}} (b_{0} − b), and γ = ϵ^{−1}W_{0}^{−1} − W^{−1} − W |b − b_{0}|^{2}. Now we work in Gibbons–Hawking coordinates using the fact that α, β and γ are invariant under the action x_{1} ↦ x_{1} + t. So if we bound the C^{k} norm of α, β and γ as functions on U_{r_{1},r_{2}} × R/ϵZ, with coordinates y and u, we can apply (2.5) to bound the C^{k} norms of α, β, γ with respect to the coordinates x and y. The interpolation inequalities then give C^{k′,α′} bounds for any k′ < k.

First look at α. Now

V = V_{0} + \frac{1}{4πϵ} log |y|^{2} + ϵ^{−1} Im(τ_{2})
= ϵ^{−1} Im(τ_{2}) + g(u, y)
= ϵ^{−1}W_{0}^{−1} + g(u, y),

where g(u, y) is, by Lemma 3.1 (c), a harmonic function on U_{r_{1},r_{2}}×R/ϵZ with ∥g∥_{C^{0}} being O(e^{−C/ϵ}). It then follows from [11], Theorem 2.10, that for each k, ∥g∥_{C^{k}} is also O(e^{−C/ϵ}). Thus

α = ϵW_{0}^{−1} − W W_{0}^{−2}

= \frac{ϵ}{W_{0}} − \frac{ϵ}{W_{0} + ϵgW_{0}^{2}}

= \frac{ϵ^{2}gW_{0}^{2}}{W_{0}(W_{0} + ϵgW_{0}^{2})}.

Now using the fact that the denominator is bounded above and below, and observing that any derivative of α will have, in the numerator, only terms which include factors of g or its derivatives, we see that for each k, ∥α∥_{C^{k}} is O(e^{−C/ϵ}).

Next look at β. By construction,

0 = ∫_{X_{b}} βdx_{1} ∧ dx_{2} = ∫_{X_{b}} βθ_{0} ∧ V du = ∫_{0}^{ϵ} W_{0}^{−1}(b − b_{0})du.

Thus b − b_{0}, which is a function on U_{r_{1},r_{2}} × S^{1}(ϵ) (even though b and b_{0} are not) has no constant term in its Fourier expansion. Both b and b_{0}, however, are quasi-periodic in u, i.e., consist of a linear plus periodic term. Let b̃ and b̃_{0} denote the periodic part (not including the constant term) of b and b_{0} respectively. Then b − b_{0} = b̃ − b̃_{0}, and we can bound the C^{k} norm of b̃ and b̃_{0} separately. For example, by (2.6), ∂_{u}b̃ = 2i∂_{y}g(u, y), which is O(e^{−C/ϵ}), and then the Poincaré inequality implies ∥b̃∥_{C^{0}} is O(e^{−C/ϵ}). Similar arguments apply to ∥b̃_{0}∥_{C^{0}}, using the explicit form for b_{0}, and from this one obtains O(e^{−C/ϵ}) bounds on ∥β∥_{C^{k}} for each k. Similar arguments apply for γ = −g(u, y)−W |b−b_{0}|^{2}.

Thus the last three terms of ω_{new} are O(e^{−C/ϵ}), and since the sum of the first two terms have eigenvalues O(ϵ^{−1}) and O(ϵ), it is clear that for sufficiently small ϵ, ω_{new} is positive definite. On the other hand, it is not clear that ∫_{X} ω_{new}^{2} = ∫_{X}(Re Ω)^{2}. However, by construction ω_{new}^{2} = (Re Ω)^{2} outside of f^{−1}(U_{r_{1},r_{2}}), and ω_{new}^{2} and (Re Ω)^{2} differ only by O(e^{−C/ϵ}) on f^{−1}(U_{r_{1},r_{2}}), so ∫_{X} ω_{new}^{2} − ∫_{X}(Re Ω)^{2} = O(e^{−C/ϵ}). Now noting that

([ω_{new}] + aE)^{2} = [ω_{new}]^{2} + 2aϵ,

we can find a two-form α on B supported on U_{r_{1},r_{2}} with C^{k} norm O(e^{−C/ϵ}) such that ∫_{X}(ω_{new} +f^{∗}α)^{2} = ∫_{X}(Re Ω)^{2}. Set ω_{ϵ} = ω_{new} +f^{∗}α. Because α is still small, ω_{ϵ} is still positive and defines the desired Kähler metric. Properties (1)–(4) are then satisfied by construction. Note that F_{ϵ} = log(Ω ∧ Ω̄/2ω_{ϵ}^{2}) is zero outside of f^{−1}(U_{r_{1},r_{2}}), and ω_{ϵ} is within O(e^{−C/ϵ}) of ω_{SF} on f^{−1}(U_{r_{1},r_{2}}). Thus ∥F_{ϵ}∥_{C^{0}} is O(e^{−C/ϵ}). The same is true of ∥F_{ϵ}∥_{C^{2}}, and since the coefficients of the metric are at worst O(ϵ) or O(ϵ^{−1}) in f^{−1}(U_{r_{1},r_{2}}), ∥ΔF_{ϵ}∥_{C^{0}} is also O(e^{−C/ϵ}). Furthermore, the Ricci form is i∂ ¯∂F_{ϵ}, which is O(e^{−C/ϵ}). This gives (5) and (6).

To bound the diameter of X with the metric ω_{ϵ}, first restrict the metric to the zero section σ_{0} of f. Identifying σ_{0} with the base B, we note that on B \ ⋃ U_{2}^{i}, the Kähler form of this restricted metric is \frac{i}{2}(ϵW_{0})((ϵW_{0})^{−2} + |b_{0}|^{2})dy ∧ dȳ. But b_{0} = 0 on σ_{0}, so this is just \frac{i}{2}ϵ^{−1}W_{0}^{−1}dy ∧ dȳ. Let D be the diameter of B \ ⋃ U_{2}^{i} under the metric \frac{i}{2}W_{0}^{−1}dy ∧ dȳ; this is independent of ϵ. Thus Diam(B \ ⋃ U_{2}^{i}) = Dϵ^{−1/2} under the metric induced by ω_{ϵ}. On the other hand, the diameter of each fibre over B \ ⋃ U_{2}^{i} is bounded by some constant times ϵ^{1/2}, so Diam(f^{−1}(B \ ⋃ U_{2}^{i})) ≤ D′ϵ^{−1/2} for sufficiently small ϵ. Then applying Corollary 3.7 to each f^{−1}(U_{2}^{i}), (keeping in mind that the changes to the metric in the gluing area are negligible for small ϵ), we see in fact that

Diam(X) ≤ D′ϵ^{−1/2} + D′′ϵ^{−1/2},

which we can always bound by D_{5}ϵ^{−1/2} for some constant D_{5}. Hence (7).

Finally, (8) follows immediately from Proposition 3.8, Remark 2.7, and the fact that any non-singular fibre has a neighbourhood in which ω_{ϵ} is arbitrarily close to the semi-flat metric for ϵ sufficiently small.

q.e.d.

Theorem 4.5. Let j : J → P^{1} be an elliptically fibred K3 surface with section and 24 singular fibres over Δ = {p_{1}, …, p_{24}} as above. Then there exists open sets U_{1}^{i} ⊆ U_{2}^{i} ⊆ P^{1}, i = 1, …, 24, each diffeomorphic to a disc, U_{j}^{i} ∩ Δ = {p_{i}}, positive constants D_{1}, D_{2}, D_{3}, D_{4}, D_{5} and ϵ_{0} such that, for all ϵ < ϵ_{0}, for any elliptic fibration f : X → P^{1} with Jacobian j : J → P^{1} with holomorphic 2-form Ω with [Re Ω]^{2} = [Re Ω_{J}]^{2}, and for any Kähler class [ω_{ϵ}] on X with [ω_{ϵ}].X_{b} = ϵ and [ω_{ϵ}]^{2} = [Re Ω]^{2} = [Im Ω]^{2}, there exists a Kähler metric ω_{ϵ} representing [ω_{ϵ}] on X with the following properties:

(1) ω_{ϵ}|_{f^{−1}(P^{1}\⋃_{i} U_{2}^{i})} is a semi-flat metric (not necessarily the standard one).

(2) ω_{ϵ}|_{f^{1}(U_{1}^{i})} = T_{σ_{i}}^{∗}ω_{OV} , where ω_{OV} is an Ooguri–Vafa metric and T_{σ_{i}} denotes translation by a (not necessarily holomorphic) section.

(3) If F_{ϵ} = log (\frac{Ω∧Ω̄/2}{ω_{ϵ}^{2}}), then

∥F_{ϵ}∥_{C^{0}} ≤ D_{1}e^{−D_{2}/ϵ}

and

∥ΔF_{ϵ}∥_{C^{0}} ≤ D_{1}e^{−D_{2}/ϵ},

where Δ denotes the Laplacian with respect to ω_{ϵ}.

(4)

inf_{v}{Ric(v, v) | |v|_{ω_{ϵ}} = 1} ≥ −D_{3}e^{−D_{4}/ϵ}.

(5) With the Riemannian metric induced by ω_{ϵ}, Diam(X) ≤ D_{5}ϵ^{−1/2}.

(6) If R denotes the Riemann curvature tensor, then

∥R∥_{C^{0}} ≤ D_{6}ϵ^{−1} log ϵ^{−1},
∥R∥_{C^{0}} → ∞ as ϵ → 0,

and on any non-singular fibre, there exists a constant C depending on the fibre such that

∥R∥ ≤ Cϵ.

Proof. First note that as in §1, we think of X as a K3 surface obtained from J simply by altering the holomorphic 2-form Ω_{J} on J to Ω_{J} + j^{∗}α, for some 2-form α on P^{1}. Thus it is natural to identify the underlying manifolds X and J, and we are only changing the complex structure. So we can think of [ω_{ϵ}] ∈ H^{2}(J, R), and in particular, in the notation of §1, we can write

[ω_{ϵ}] = ϵ(σ_{0} + B) mod E

for some B ∈ E^{⊥}/E ⊗ R. Furthermore, given the values of the classes [ω_{ϵ}] and [Ω] modulo E, and given that [ω_{ϵ}], [Re Ω], [Im Ω] form a hyperkähler triple, the classes [ω_{ϵ}] and [Ω] are completely determined. Thus the choice of B uniquely determines the Kähler class and complex structure.

We modify the role of B slightly. Let ω_{ϵ}^{0} be the Kähler form on J provided by Theorem 4.4. Then in fact we can write

[ω_{ϵ}] − [ω_{ϵ}^{0}] = ϵB mod E

for some B ∈ E^{⊥}/E ⊗ R. The class B still determines all data. So fix this class in E^{⊥}/E ⊗ R. This latter vector space is naturally identified with H^{1}(P^{1}, R^{1}j_{∗}R). Consider the exact sequence

0 → R^{1}j_{∗}R → C^{∞}(T_{B}^{∗}) → F → 0.

Here C^{∞}(T_{B}^{∗}) denotes the sheaf of C^{∞} sections of T_{B}^{∗}, and the first map is induced by tensoring the inclusion R^{1}j_{∗}Z ↪ T_{B}^{∗} with R. This gives a surjection H^{0}(P^{1}, F) → H^{1}(P^{1}, R^{1}j_{∗}R). Now a section of F is given by an open covering {U_{i}} of P^{1} and sections σ_{i} ∈ Γ(U_{i}, C^{∞}(T_{B}^{∗})) with σ_{i} − σ_{j} ∈ Γ(U_{i}, R^{1}j_{∗}R). This open covering {U_{i}} can always be chosen with the following properties:

(1) Each U_{i} contains at most one point of Δ, and if p_{j} ∈ U_{i}, then \overline{U_{2}^{j}} ⊆ U_{i}.

(2) Each U_{i} is convex with respect to some metric on P^{1}, so that all multiple intersections of the U_{i}’s are contractible.

(3) If U_{i} ∩ Δ = ϕ, then U_{i} ∩ ⋃_{j} U_{2}^{j} = ϕ.

In fact, fixing one such open covering, all sections of F can be represented over this open covering.

Now represent B by (U_{i}, σ_{i}), and let T_{σ_{i}} : f^{−1}(U_{i}) → f^{−1}(U_{i}) denote translation by the section σ_{i}. Now consider the forms T_{σ_{i}}^{∗}ω_{ϵ}^{0} and T_{σ_{i}}^{∗}Ω_{J}. On f^{−1}(U_{i}∩U_{j}), ω_{ϵ}^{0} is the standard semi-flat metric, by condition (3) on the open covering above, and since σ_{j} −σ_{i} is a flat section with respect to the Gauss-Manin connection, T_{σ_{j}−σ_{i}} is an isometry, i.e., T_{σ_{j}−σ_{i}}^{∗}ω_{ϵ}^{0} = ω_{ϵ}^{0}, T_{σ_{j}−σ_{i}}^{∗}Ω_{J} = Ω_{J} (see Example 2.2). Thus

T_{σ_{i}}^{∗}ω_{ϵ}^{0} = T_{σ_{i}}^{∗}T_{σ_{j}−σ_{i}}^{∗}ω_{ϵ}^{0} = T_{σ_{j}}^{∗}ω_{ϵ}^{0},

and similarly T_{σ_{i}}^{∗}Ω_{J} = T_{σ_{j}}^{∗}Ω_{J}. Thus these forms glue, to give global forms ω_{ϵ}, Ω on the manifold J. The 2-form Ω satisfies Ω ∧ Ω = 0, and thus induces a new complex structure on J. An easy local calculation shows that Ω = Ω_{J} + j^{∗}α′, for some 2-form α′ on P^{1}. Furthermore, it is clear that

∫_{J_{b}} ω_{ϵ} = ϵ, ∫_{J} ω_{ϵ} ∧ Ω = 0, and ∫_{J} ω_{ϵ}^{2} = ∫_{J}(Re Ω)^{2} = ∫_{J}(Im Ω)^{2}.

Thus the cohomology classes [ω_{ϵ}], [Re Ω] and [Im Ω] form a hyperkähler triple. If we show that

[ω_{ϵ}] − [ω_{ϵ}^{0}] = ϵB mod E,

then we have constructed a Kähler form in the desired class (deducing moreover that the new complex structure is just that obtained from X).

To see the required identity, observe that we have an exact sequence

H_{f^{−1}(Δ)}^{2}(X, R)⟶^{φ}H^{2}(X, R)⟶H^{2}(X_{0}, R)⟶H_{f^{−1}(Δ)}^{3}(X, R).

Now H_{f^{−1}(Δ)}^{2}(X, R) = H^{0}(f^{−1}(Δ), R) = R^{24}, and the image of φ is just the one-dimensional subspace of H^{2}(X, R) spanned by [E], the class of a fibre. Thus it is enough to show that

[ω_{ϵ}|_{X_{0}}] − [ω_{ϵ}^{0}|_{X_{0}}] = ϵB ∈ H^{2}(X, R)/E ⊆ H^{2}(X_{0}, R).

Now on X_{0}, ω_{ϵ}^{0} is cohomologous to ω_{SF} by construction, and ω_{ϵ} is cohomologous to a Kähler form ω′_{SF} obtained in the same way as ω_{ϵ} via translation and gluing, but starting from ω_{SF} rather than ω_{ϵ}^{0}. Thus it is enough to show that on X_{0}

[ω′_{SF}] − [ω_{SF}] = ϵB.

Over an open set U_{i}, write

ω_{SF} = \frac{i}{2}W_{0}^{−1}(ϵ ϑ_{v} ∧ ϑ̄_{v} + ϵ^{−1} ϑ_{h} ∧ ϑ̄_{h}).

Now

T_{σ_{i}}^{∗}(ϑ_{v}) = ϑ_{v} + W_{0}(∂_{y}σ_{i} + b_{0}(σ_{i}(y), y)) ϑ_{h} + W_{0}(∂ȳσ) ϑ̄_{h},

so over U_{i}

ω′_{SF} − ω_{SF} =T_{σ_{i}}^{∗}ω_{SF} − ω_{SF}
=\frac{ϵi}{2}((∂_{y}σ_{i} + b_{0}(σ_{i}(y), y)) ϑ_{h} ∧ ϑ̄_{v} + ∂ȳσ_{i} ϑ̄_{h} ∧ ϑ̄_{v}
+ ∂_{y}σ̄_{i} ϑ_{v} ∧ ϑ_{h}
+ (∂ȳσ̄_{i} + b̄_{0}(σ_{i}(y), y)) ϑ_{v} ∧ ϑ̄_{h}) mod ϑ_{h} ∧ ϑ̄_{h}
=\frac{ϵi}{2}d(σ_{i}ϑ̄_{v} − σ̄_{i}ϑ_{v}) mod ϑ_{h} ∧ ϑ̄_{h}

as can be easily seen using (4.1) and a calculation similar to that of (4.2).

How does the two-form ω′_{SF} − ω_{SF} determine an element of H^{1}(B_{0}, R^{1}f_{0∗}R)? Given the open covering {U_{i}} of B_{0}, if ω′_{SF} − ω_{SF} is an exact form on each f^{−1}(U_{i}), we can write ω′_{SF} − ω_{SF} = dα_{i} for some 1-form α_{i} on f^{−1}(U_{i}). Then on f^{−1}(U_{i}∩U_{j}), α_{i}−α_{j} is closed, and hence determines an element of H^{1}(f^{−1}(U_{i} ∩ U_{j}), R) = Γ(U_{i} ∩ U_{j}, R^{1}f_{0∗}R) for our choice of open covering. Now we have found such α_{i} modulo ϑ_{h}∧ϑ̄_{h}, so ω′_{SF} − ω_{SF} is represented by a Čech cocycle for R^{1}f_{0∗}R given by

(U_{i} ∩ U_{j}, \frac{ϵi}{2}((σ_{i} − σ_{j})ϑ̄_{v} − (σ̄_{i} − σ̄_{j})ϑ_{v})).

By integrating this one-form over the periods, one sees this is precisely the section of Γ(U_{i} ∩ U_{j}, R^{1}f_{0∗}R) given by ϵ(σ_{i} − σ_{j}). Thus ω′_{SF} − ω_{SF} represents the class ϵB.

Finally, properties (1)–(4) and (6) follow immediately from Theorem 4.4, (3)–(6) and (8). On the other hand, the diameter of f^{−1}(U_{i}) with respect to ω_{ϵ} is the same as the diameter of f^{−1}(U_{i}) with respect to the metric ω_{ϵ}^{0}. Since there are a fixed number of U_{i}’s, the estimate on the diameter continues to hold from Theorem 4.4, (7). q.e.d.

Remark 4.6. In the construction of the proof of Theorem 4.5, we may sometimes want to be able to control the sections σ_{i} we use to represent the class B. This can be done as follows. The class of B depends on the choice of the zero section σ_{0}. Changing the class of the zero section changes B by an element of E^{⊥}/E. Thus B really should be thought of as living in E^{⊥}/E ⊗R/Z. (See [13] or [14], §7.) Thus in some cases we might want to choose a compact set F in E^{⊥}/E ⊗R containing a fundamental domain for E^{⊥}/E. We can then choose for each B ∈ F a representative (σ_{i}) of B with various norms, as required, bounded by constants independent of B ∈ F. We will say we are choosing the B-field B in a fundamental domain for the B-field.

5. Ricci-flat metrics

We will continue with the setting of Theorem 4.5. In other words, we have a fixed Jacobian elliptic fibration j : J → P^{1}. Our goal is to show that there exists an ϵ_{0} such that for any f : X → P^{1} with Jacobian j : J → P^{1}, and any ϵ < ϵ_{0}, and any metric ω_{ϵ} given by Theorem 4.5, there exists a function u_{ϵ} such that ω_{ϵ} + i∂ ¯∂u_{ϵ} is a Ricciflat metric, and furthermore that u_{ϵ} is very small in the C^{k,α} sense. Of course, that such a u_{ϵ} exists is Yau’s proof of the Calabi conjecture. Here we apply standard techniques, following [20], to obtain control of u_{ϵ}. As mentioned in the introduction, the only subtle difference is that as ϵ → 0, Diam(X) → ∞, and this requires us to be a bit more careful in estimating constants. However, we follow [20] closely.

More precisely, we wish to solve the equations

\begin{aligned}(ω_{ϵ} + i∂ ¯∂u_{ϵ})^{2} & = e^{F_{ϵ}}ω_{ϵ}^{2} \\ ∫_{X} u_{ϵ}ω_{ϵ}^{2} & = 0. \end{aligned} (5.1)

Here F_{ϵ} = log(\frac{Ω∧Ω̄/2}{ω_{ϵ}^{2}}). By [35], we know such a u_{ϵ} exists.

We begin with some standard lemmas. For convenience, we will assume V ol(X) = 1. This can be achieved since we are holding the volume of X constant anyway, so we just scale the original Ω so that ∫_{J}(Re Ω)^{2} = 1.

Lemma 5.1. Let X, ω_{ϵ} be as in Theorem 4.5. Assume V ol(X) = 1. Then there exists a function I(ϵ) depending only on ϵ and J with I(ϵ) ≥ Cϵ^{5}, C depending only on J, such that:

(1) For any function f on X such that ∫_{X} fω_{ϵ}^{2} = 0,

∥df∥_{2}^{2} ≥ I(ϵ)∥f∥_{4}^{2}.

(2) For any function f on X,

∥df∥_{2}^{2} ≥ I(ϵ)(∥f∥_{4}^{2} − ∥f∥_{2}^{2}).

Proof. These are the standard Sobolev inequalities, but we just need to be careful about the constants. We have, by [23], Lemmas 1 and 2, for a function f such that ∫_{X} fω_{ϵ}^{2} = 0,

∥df∥_{2}^{2} ≥ C_{2}∥f∥_{4}^{2}

while for an arbitrary function, we have

∥df∥_{2}^{2} ≥ D(4)C_{2}(∥f∥_{4}^{2} − ∥f∥_{2}^{2}).

Here, we are using Li’s notation for the constants C_{0}, C_{1}, C_{2}, D(n) and the fact that the volume is 1 and the dimension is 4. Again by [23], D(4) is an absolute constant, C_{2} = D(4)C_{0}^{1/2}, and 2C_{1} ≥ C_{0} ≥ C_{1}, where C_{1} is the constant in the isoperimetric inequality

C_{1}(min{V (M_{1}), V (M_{2})})^{3} ≤ V (N)^{4}

where V denotes volume, and N is any codimension one submanifold of X dividing it into M_{1} and M_{2}. In [9], Croke calls this constant Φ(M).

Theorem 13 from [9] says that

C_{1} ≥ C_{4} (∫_{0}^{Diam(X)}((\sqrt{1/K}) sinh(\sqrt{K}r))^{3}dr)^{−5} ,

where C_{4} again is an absolute constant, and Ric(X) ≥ −3K, where 3K ≤ D_{3}e^{−D_{4}/ϵ} by Theorem 4.5, (4). Now the integral is bounded above by

Diam(X)(\sqrt{1/K} sinh(\sqrt{K}Diam(X)))^{3}.

Now by Theorem 4.5, (5), \sqrt{K}Diam(X) → 0 as ϵ → 0, so for sufficiently small ϵ, using the first term of the Taylor series expansion of sinh, this is bounded by

C_{5}Diam(X)^{4} ≤ C_{6}ϵ^{−2}

so C_{1} ≥ C_{7}ϵ^{10}, hence C_{0} ≥ C_{8}ϵ^{10} and we can take

I(ϵ) = min(D(4), 1)C_{2} ≥ C_{9}ϵ^{5}. q.e.d.

Lemma 5.2. (The C^{0} estimate.) Let u_{ϵ} be the solution to Equations (5.1). There exists a constant C depending only on J, such that for all ϵ < ϵ_{0}, (ϵ_{0}, D_{2} as in Theorem 4.5)

∥u_{ϵ}∥_{∞} ≤ Cϵ^{−5}e^{−D_{2}/ϵ}.

Proof. The starting point is the inequality (23) of [20]:

∫_{X} |d|u_{ϵ}|^{p/2}|^{2} ≤ Ap ∫_{X} |F_{ϵ}||u_{ϵ}|^{p−1}.

All integrals are with measure ω_{ϵ}^{2}. See also the expanded derivation of this inequality in [24]. One can check the constant A is independent of p and ϵ.

We apply this first with p = 2. The left-hand side is ∥du_{ϵ}∥_{2}^{2} ≥ A_{1}ϵ^{5}∥u_{ϵ}∥_{4}^{2} by Lemma 5.1, (1), so by applying Hölder’s inequality to the right-hand side, we get

(∫_{X} |u_{ϵ}|^{4})^{1/2} ≤ A_{2}ϵ^{−5} (∫_{X} |F_{ϵ}|^{4/3})^{3/4} (∫_{X} |u_{ϵ}|^{4})^{1/4}

or

\begin{aligned}∥u_{ϵ}∥_{4} & ≤ A_{3}ϵ^{−5} (∫_{X} |F_{ϵ}|^{4/3})^{3/4} \\ & ≤ C_{1}ϵ^{−5}e^{−D_{2}/ϵ}. \end{aligned} (5.2)

Now for arbitrary p, using Lemma 5.1, (2)

∥u_{ϵ}∥_{2p}^{p} = (∫_{X} |u_{ϵ}^{2p}|)^{1/2} = ∥ |u_{ϵ}|^{p/2}∥_{4}^{2}
≤ A_{4}ϵ^{−5}∥ d|u_{ϵ}|^{p/2}∥_{2}^{2} + ∥ |u_{ϵ}|^{p/2}∥_{2}^{2}
≤ A_{5}pϵ^{−5} (∫_{X} |F_{ϵ}| |u_{ϵ}|^{p−1}) + ∥ |u_{ϵ}|^{p/2}∥_{2}^{2}.

Applying Hölder’s inequality to the first term, we have, with q = p, q′ = 1/(1 − 1/p) = p/(p − 1),

∫_{X} |F_{ϵ}| |u_{ϵ}|^{p−1} ≤ ∥F_{ϵ}∥_{p} ∥ |u_{ϵ}|^{p−1}∥_{p/(p−1)}
= ∥F_{ϵ}∥_{p} ∥u_{ϵ}∥_{p}^{p−1},

so

\begin{aligned}∥u_{ϵ}∥_{2p}^{p} & ≤ A_{5}pϵ^{−5}∥F_{ϵ}∥_{p} ∥u_{ϵ}∥_{p}^{p−1} + ∥u_{ϵ}∥_{p}^{p} \\ & = (A_{5}pϵ^{−5}∥F_{ϵ}∥_{p} + ∥u_{ϵ}∥_{p}) ∥u_{ϵ}∥_{p}^{p−1}. \end{aligned} (5.3)

Now we claim that if we set p_{n} = 2^{n+1}, there exists constants C_{n} such that

∥u_{ϵ}∥_{p_{n}} ≤ C_{n}ϵ^{−5}e^{−D_{2}/ϵ}

for all ϵ < ϵ_{0}. This holds for n = 1 by (5.2). Suppose it holds for a given n. Then by (5.3),

∥u_{ϵ}∥_{p_{n+1}}^{p_{n}} ≤ (A_{5}p_{n}ϵ^{−5}D_{1}e^{−D_{2}/ϵ} + C_{n}ϵ^{−5}e^{−D_{2}/ϵ})(C_{n}ϵ^{−5}e^{−D_{2}/ϵ})^{p_{n}−1}

≤ \begin{cases}(A_{5}D_{1}2^{n+1} + 1)(C_{n}ϵ^{−5}e^{−D_{2}/ϵ})^{p_{n}} & if C_{n} ≥ 1; \\ (A_{5}D_{1}2^{n+1} + 1)(ϵ^{−5}e^{−D_{2}/ϵ})^{p_{n}} & if C_{n} ≤ 1. \end{cases}

Thus we can take

C_{n+1} ≤ \begin{cases}(A_{5}D_{1}2^{n+1} + 1)^{2^{−(n+1)}}C_{n} & if C_{n} ≥ 1; \\ (A_{5}D_{1}2^{n+1} + 1)^{2^{−(n+1)}} & if C_{n} ≤ 1. \end{cases}

It then follows as in [20], page 299, that C_{n} ≤ A_{6} for some constant A_{6} independent of n and ϵ. Thus we conclude that

∥u_{ϵ}∥_{∞} ≤ A_{6}ϵ^{−5}e^{−D_{2}/ϵ}

for all ϵ < ϵ_{0}. q.e.d.

Lemma 5.3. (The C^{2} estimate.) Let u_{ϵ} be the solution to Equations (5.1). There are constants C and ϵ_{0} depending only on J (possibly smaller than the ϵ_{0} of Theorem 4.5) such that for all ϵ < ϵ_{0},

C^{−1}ω_{ϵ} ≤ ω̃_{ϵ} ≤ Cω_{ϵ}

where ω̃_{ϵ} = ω_{ϵ} + i∂ ¯∂u_{ϵ}.

Proof. Let R_{ϵ} = sup_{i≠j} |R_{iījj̄}|, where R_{iījj̄} is the holomorphic bisectional curvature of the metric ω_{ϵ}, and the supremum is over all points of X and unitary bases at each point. Since the holomorphic bisectional curvature determines the curvature, ([6], pg. 76) and sup ∥R∥ → ∞ as ϵ → 0 by Theorem 4.5 (6), we must have R_{ϵ} > 1 for small ϵ. So if c_{ϵ} = 2R_{ϵ}, then c_{ϵ} + inf R_{iījj̄} ≥ R_{ϵ} > 1. Here the infinum is as before over all unitary frames and points on X. Then [35], (2.22), reads

e^{c_{ϵ}u_{ϵ}}Δ′(e^{−c_{ϵ}u_{ϵ}}(2 + Δu_{ϵ})) ≥ (ΔF_{ϵ} − 4 inf_{i≠j} R_{iījj̄}(x))

− 2c_{ϵ}(2 + Δu_{ϵ})

+ (c_{ϵ} + inf_{i≠j} R_{iījj̄}(x)) e^{−F_{ϵ}} (2 + Δu_{ϵ})^{2}

where the infina are now only at the given point (but still over all unitary bases). Here Δ′ is the Laplacian with respect to the metric ω_{ϵ} + i∂ ¯∂u_{ϵ}, and Δ is the Laplacian with respect to ω_{ϵ}. Let

k(x) = − inf_{i≠j} R_{iījj̄}(x)/R_{ϵ},

so that |k(x)| ≤ 1.

Now suppose e^{−c_{ϵ}u_{ϵ}}(2+Δu_{ϵ}) assumes its maximum at x ∈ X. Then by the maximum principal, the Laplacian must be non-positive there, so at the point x

0 ≥e^{c_{ϵ}u_{ϵ}}Δ′(e^{−c_{ϵ}u_{ϵ}}(2 + Δu_{ϵ}))
≥(ΔF_{ϵ} + 4k(x)R_{ϵ}) − 2c_{ϵ}(2 + Δu_{ϵ}) + (c_{ϵ} − k(x)R_{ϵ})e^{−F_{ϵ}}(2 + Δu_{ϵ})^{2}
=(ΔF_{ϵ} + 4k(x)R_{ϵ}) − 4R_{ϵ}(2 + Δu_{ϵ}) + (2 − k(x))R_{ϵ}e^{−F_{ϵ}}(2 + Δu_{ϵ})^{2}
=e^{−F_{ϵ}}(2 − k(x))R_{ϵ} [((2 + Δu_{ϵ}) − \frac{2e^{F_{ϵ}}}{2 − k(x)})^{2} − (\frac{2e^{F_{ϵ}}}{2 − k(x)})^{2}

+\frac{e^{F_{ϵ}}(ΔF_{ϵ} + 4R_{ϵ}k(x))}{(2 − k(x))R_{ϵ}}]

and since |k(x)| ≤ 1, we get

|(2 + Δu_{ϵ}) − \frac{2e^{F_{ϵ}}}{2 − k(x)}| ≤ |(\frac{2e^{F_{ϵ}}}{2 − k(x)})^{2} − \frac{e^{F_{ϵ}}(ΔF_{ϵ} + 4R_{ϵ}k(x))}{(2 − k(x))R_{ϵ}} |^{1/2} .

If we are outside of the region where the gluing is taking place, then F_{ϵ} = 0, so we get

|(2 + Δu_{ϵ}) − \frac{2}{2 − k(x)}| ≤ |(\frac{2}{2 − k(x)})^{2} − \frac{4k(x)}{2 − k(x)}|^{1/2} ,

or

2 + Δu_{ϵ} ≤ \frac{2}{2 − k(x)} + |(\frac{2}{2 − k(x)})^{2} − \frac{4k(x)}{2 − k(x)}|^{1/2}
= 2.

In the gluing region, by Theorem 4.5 (6), there is a constant C_{1}

|k(x)| ≤ C_{1}ϵ/R_{ϵ}.

Also in the gluing region, we can use the bounds of Theorem 4.5,

(3) on F_{ϵ} and ΔF_{ϵ}, to get, for a constant C_{2} bounding e^{F_{ϵ}},

2 + Δu_{ϵ} ≤ \frac{2e^{F_{ϵ}}}{2 − k(x)} + |(\frac{2e^{F_{ϵ}}}{2 − k(x)})^{2} − \frac{e^{F_{ϵ}}(ΔF_{ϵ} + 4R_{ϵ}k(x)))}{(2 − k(x))R_{ϵ}} |^{1/2}

≤ \frac{2C_{2}}{2 − C_{1}ϵ/R_{ϵ}} + |(\frac{2C_{2}}{2 − C_{1}ϵ/R_{ϵ}} )^{2} + \frac{C_{2}(D_{1}e^{−D_{2}/ϵ} + 4C_{1}ϵ)}{2 − C_{1}ϵ/R_{ϵ}} |^{1/2} .

Now as ϵ → 0, ϵ/R_{ϵ} → 0 by Theorem 4.5, (6). So what we get is

(2 + Δu_{ϵ})(x) ≤ C_{3}

for sufficiently small ϵ, and C_{3} independent of ϵ.

Now

e^{−c_{ϵ}u_{ϵ}(y)}(2 + Δu_{ϵ})(y) ≤ e^{−c_{ϵ}u_{ϵ}(x)}(2 + Δu_{ϵ})(x)

for all points y, so

2 + Δu_{ϵ} ≤ e^{c_{ϵ}(u_{ϵ}(y)−u_{ϵ}(x))}C_{3}
≤ e^{c_{ϵ}(sup u_{ϵ}−inf u_{ϵ})}C_{3}
≤ e^{R_{ϵ}C_{4}ϵ^{−5}e^{−D_{2}/ϵ}}C_{3}.

By Theorem 4.5, (6),

R_{ϵ}ϵ^{−5}e^{−D_{2}/ϵ} → 0,

so we get

2 + Δu_{ϵ} ≤ C_{5}

for sufficiently small ϵ.

Now working in a choice of coordinates z_{1}, z_{2} at a point so that ∂_{z_{1}}, ∂_{z_{2}} are unitary at the point with respect to ω_{ϵ} and which also diagonalizes ω̃_{ϵ} = ω_{ϵ} + i∂ ¯∂u_{ϵ}, then

(ω̃_{ϵ})_{ij̄} = δ_{ij}(1 + (u_{ϵ})_{iī}),

and each 1 + u_{iī} is positive, so 1 + (u_{ϵ})_{iī} ≤ C_{5}, so ω̃_{ϵ} ≤ C_{5}ω_{ϵ}. Also,

ω̃_{ϵ}^{2} = ∏(1 + (u_{ϵ})_{iī}) ω_{ϵ}^{2} = e^{F_{ϵ}}ω_{ϵ}^{2}.

Since 1 + (u_{ϵ})_{iī} is bounded above by C_{5}, it must be bounded below by something close to C_{5}^{−1}, so changing C_{5} slightly if necessary, we get

C_{5}^{−1}ω_{ϵ} ≤ ω̃_{ϵ} ≤ C_{5}ω_{ϵ}. q.e.d.

We note here that for some purposes, Lemma 5.3 is already sufficient. For example, if we wish to know that the fibres collapse to points as ϵ → 0, Lemma 5.3 along with Proposition 3.5 tells us the diameter of each fibre under the Ricci-flat metric goes to zero as ϵ → 0. However, if we wish to get a clearer picture of the asymptotic behaviour of the metric, we need stronger results.

Lemma 5.4. (The C^{2,α} estimate.) Let u_{ϵ} be the solution to Equations (5.1). If U ⊆ B is a simply connected open set with \overline{U} ⊆ B_{0} = B \Δ, then there exists constants α and ϵ_{0} and a polynomial P, depending on J and U, such that

∥u_{ϵ}∥_{C^{2,α}} ≤ P (ϵ^{−1})

in f^{−1}(U) for all ϵ < ϵ_{0} and B in a fundamental domain for the Bfield (see Remark 4.6). Here the C^{2,α} norm is on f^{−1}(U) as defined in Lemma 4.1, and so α, ϵ_{0} and P also depend on the choice of holomorphic coordinate y and fixed bounded domain T′, as specified in the proof below.

Proof. We need to apply the basic result of [11], Theorem 17.14. However, we must be careful about the constants. Let π : T_{B}^{∗} → B be the projection, and let T′ ⊆ π^{−1}(U) ⊆ T_{B}^{∗} be a fixed bounded domain which contains a fundamental domain of each fibre of f over U. We will be computing norms in the domain T′. To do so, we choose a holomorphic coordinate y in the base, yielding holomorphic canonical coordinates x, y on T_{U}^{∗}. Now take a bigger open set T (ϵ) containing T′. This open set will also be bounded, but will depend on ϵ. We choose it as follows. First let V ⊆ B_{0} be an open set with \overline{U} ⊆ V, \overline{V} ⊆ B_{0}, and the holomorphic coordinate y extending to V. Let T ⊆ π^{−1}(V) be a domain containing T′ and containing a fundamental domain of each fibre over V. Let

T (ϵ) = {(x, y) ∈ T_{V}^{∗} | there exists (x̃, y) ∈ T with |x − x̃| < ϵ^{−1/2}}.

The point of this choice is as follows. Consider the change of variable y′ = ϵ^{−1/2}y, x′ = ϵ^{1/2}x. Then using x′, y′ to identify T_{U}^{∗} with a subset of C^{2}, we get Dist(∂\overline{T (ϵ)}, T′) ≥ 1, for sufficiently small ϵ, in the euclidean distance in C^{2}.

Pulling back ω_{ϵ} to T (ϵ), we can write

ω_{ϵ} = i∂ ¯∂(φ_{1} + φ_{2})

where i∂ ¯∂φ_{1} is a semi-flat metric and i∂ ¯∂φ_{2} is the correction to this metric resulting from the gluing process. By applying Lemma 4.1, we can choose ϵ sufficiently small so that the C^{2,α} norm of φ_{2} on T′ is as small as we like (and the C^{2} norm of φ_{2} on T (ϵ)). On the other hand, φ_{1} can be taken to be a translation of the Kähler potential given for the standard semi-flat metric in Example 2.2. Since we have chosen B in a fundamental domain, we can then bound the C^{2,α} norm of φ_{1} on T′ independently of B as a polynomial in ϵ^{−1}. The same is true of the C^{2} norm of φ_{1} on T′(ϵ).

Now the equation that u_{ϵ} satisfies is

(i∂ ¯∂ψ_{ϵ})^{2} = Ω ∧ Ω̄/2

where ψ_{ϵ} = φ_{1} + φ_{2} + u_{ϵ}. Thus a C^{2,α} bound on ψ_{ϵ} polynomial in ϵ^{−1} yields a C^{2,α} bound on u_{ϵ} polynomial in ϵ^{−1}. Now changing coordinates between x, y and x′, y′ also only affects the C^{2,α} norm of a function by a factor polynomial in ϵ^{−1}, so we can work with respect to the coordinates x′, y′. Now in these coordinates,

i∂ ¯∂(φ_{1}) = \frac{i}{2}(W_{0}(dx′ + ϵbdy′) ∧ \overline{(dx′ + ϵbdy′)} + W_{0}^{−1}dy′ ∧ dȳ′).

By looking at the explicit form of b for the semi-flat metric, we see ϵb in fact goes to zero as ϵ → 0 on T (ϵ). Thus the eigenvalues of i∂ ¯∂φ_{1} on T (ϵ), i.e., the eigenvalues of the matrix

(\begin{matrix}W_{0} & ϵbW_{0} \\ ϵb̄W_{0} & W_{0}^{−1} + ϵ^{2}|b|^{2} \end{matrix}) ,

can be bounded below and above by some constants λ and Λ independently of ϵ. Since φ_{2} is small, the same is true of ω_{ϵ} on T (ϵ). Finally, by Lemma 5.3, the eigenvalues of i∂ ¯∂ψ_{ϵ} are bounded below and above by C^{−1}λ and CΛ, independently of ϵ. Furthermore, Lemmas 5.2 and 5.3 imply the C^{2} norm of ψ_{ϵ} on T (ϵ) is bounded by a polynomial in ϵ^{−1}. We can now apply [11], Theorem 17.14 to the domains T′ ⊆ T (ϵ), to obtain the desired result. q.e.d.

We shall now follow the standard method of continuity from [35], and, for t ∈ [0, 1], look at the solution u_{ϵ,t} to the equation

(5.4)  (ω_{ϵ} + i∂ ¯∂u_{ϵ,t})^{2} = (1 + t(e^{F_{ϵ}} − 1))ω_{ϵ}^{2},

(5.5)  ∫_{X} u_{ϵ,t} ω_{ϵ}^{2} = 0.

We set ω_{ϵ,t} = ω_{ϵ} + i∂ ¯∂u_{ϵ,t}, the Kähler form of a metric on the given complex manifold X. For t = 0, we just get back our original (glued) metric, whilst t = 1 is the case we have just looked at, yielding the Ricci flat metric with Kähler form ω̃_{ϵ}. Since log(1 + t(e^{F_{ϵ}} − 1)) has the same properties as F_{ϵ} for t ∈ [0, 1], all the above estimates of Lemmas 5.2–5.4 work equally well for u_{ϵ,t}. In particular,

C^{−1}ω_{ϵ} ≤ ω_{ϵ,t} ≤ Cω_{ϵ}

for some constant C independent of t ∈ [0, 1] and ϵ, and

∥u_{ϵ,t}∥_{C^{2,α}} ≤ P (ϵ^{−1}),

with the polynomial P independent of t ∈ [0, 1] and ϵ.

Moreover, the Ricci form of the metric ω_{ϵ,t} is given by

\frac{i}{2π} ∂ ¯∂(F_{ϵ} − log(1 + t(e^{F_{ϵ}} − 1))),

and so the Ricci curvature Ric_{ω_{ϵ,t}} has a similar lower bound (independent of t) as Ric_{ω_{ϵ}}.

Lemma 5.5. Let G_{ϵ,t}(x, y) denote Green’s function for the Laplacian Δ_{ϵ,t} associated to the metric ω_{ϵ,t}, normalised so that

∫_{X} G_{ϵ,t}(x, y) ω_{ϵ,t}^{2}(x) = 0.

Then, for ϵ sufficiently small and any t ∈ [0, 1],

G_{ϵ,t}(x, y) ≥ −Aϵ^{−11},

for some constant A independent of ϵ and t.

Proof. For ease of notation, we drop the suffices ϵ, t. We follow the proof of Lemma 3.3 from [24], which is due to Peter Li. The volume of X is 1, and we set K(x, y, s) = H(x, y, s) − 1, where H is the heat kernel on X. As in [24], we need to find a lower bound for the integral of K(x, y, s) over 1 ≤ s ≤ ∞, of the same form as that claimed for G(x, y). Lu observes that

K(x, y, s) ≥ −K^{1/2}(x, x, s)K^{1/2}(y, y, s),

and that furthermore, for any x ∈ X,

K(x, x, s) ≤ K(x, x, 1)e^{−λ(s−1)},

for all s ≥ 1, where λ denotes the first (positive) eigenvalue of the Laplacian. If now we can suitably bound λ from below, and K(x, x, 1) from above, we’ll be able to integrate the resulting function which bounds K^{1/2}(x, x, s)K^{1/2}(y, y, s) from above, obtaining a lower bound for ∫_{1}^{∞} K(x, y, s)ds.

The bound from below for λ comes from Theorem 4 on page 116 of [31]. Since the metric is within a fixed constant factor of our original metric, all the quantities in the given formula are known, and so using Theorem 4.5, and we deduce that

λ ≥ A_{1}Diam(X)^{−2} ≥ A_{2}ϵ,

for appropriate absolute constants A_{1}, A_{2}. The proof of Lemma 5.1 may be applied to the metric ω_{ϵ,t} to obtain a similar bound on the Sobolev constant, and then the bound from above for K(x, x, 1) is implied by Equation (3.12) of [36], where the argument given there has been run for the function K(x, y, s) = H(x, y, s) − 1 (so in particular ∫_{X} K(x, z, s) ω_{ϵ,t}^{2}(z) = 0). For an appropriate constant A_{3} independent of t, we have

K(x, x, 1) ≤ A_{3}ϵ^{−10}.

Thus for all s ≥ 1

K(x, x, s) ≤ A_{3}ϵ^{−10}e^{−A_{2}ϵ(s−1)},

which then implies that

K(x, y, s) ≥ −A_{4}ϵ^{−10}e^{−A_{2}ϵ(s−1)},

for some constant A_{4} independent of ϵ and t. On integrating, we obtain the claimed bound in the form stated (a rather more involved argument in fact gives a bound K(x, x, 1) ≤ A′_{3}ϵ^{−3}, and hence G_{ϵ,t}(x, y) ≥ −A′ϵ^{−4}, but this extra accuracy is not required). q.e.d.

We are now ready for our main theorem.

Theorem 5.6. For any simply connected open set U ⊆ B_{0} with \overline{U} ⊆ B_{0}, and any k ≥ 2, 0 < α < 1, there exists constants C, C′, and ϵ_{0} such that for all choices of B in a fundamental domain for the Bfield and any ϵ < ϵ_{0} giving ω_{ϵ} as in Theorem 4.5, and u_{ϵ} satisfying the equations

(ω_{ϵ} + i∂ ¯∂u_{ϵ})^{2} = e^{F_{ϵ}}ω_{ϵ}^{2}
∫_{X} u_{ϵ}ω_{ϵ}^{2} = 0

with F_{ϵ} = log(\frac{Ω∧Ω̄/2}{ω_{ϵ}^{2}}), we have

∥u_{ϵ}∥_{C^{k,α}} ≤ Ce^{−C′/ϵ}.

Here, the norm is as in Lemma 4.1 on the region f^{−1}(U), and the constants C, C′ are independent of ϵ.

Proof. This is now completely standard. Following [20] and [24], we differentiate (5.4) with respect to t, getting

Δ_{ϵ,t} \frac{du_{ϵ,t}}{dt} = \frac{e^{F_{ϵ}} − 1}{1 + t(e^{F_{ϵ}} − 1)}.

The right hand side is very small, which along with the estimate of Lemma 5.5, allows us to bound du_{ϵ,t}/dt. Indeed, by Green’s formula and (5.5), we have

\frac{du_{ϵ,t}(x)}{dt} = − ∫_{X} (Δ_{ϵ,t} \frac{du_{ϵ,t}}{dt} ) G̃_{ϵ,t}(x, y)ω_{ϵ,t}^{2}(y).

Here G̃_{ϵ,t} is the Green’s function for the Laplacian for ω_{ϵ,t}, normalized so that inf_{X} G̃_{ϵ,t} = 0. Lemma 5.5 tells us that ∫_{X} G̃_{ϵ,t}(x, y)ω_{ϵ,t}^{2}(y) ≤ Aϵ^{−11} for some constant A independent of ϵ and t, so bounds on F_{ϵ} imply

(5.6)  ∥du_{ϵ,t}/dt∥_{C^{0}} ≤ C_{1}e^{−C_{2}/ϵ}

for some constants C_{1} and C_{2} independent of t and ϵ, for sufficiently small ϵ.

We can now apply the interior Schauder estimates (see [11] Theorem 6.2) to obtain

(5.7)  ∥du_{ϵ,t}/dt∥_{C^{2,α}} ≤ C_{3}e^{−C_{4}/ϵ}

for sufficiently small ϵ. This holds for α as given by Lemma 5.4. We note that a certain amount of care must be taken in applying these estimates: first, we need to use the estimate of (5.6) and Lemma 5.4 on a larger open set U′ with \overline{U} ⊆ U′ ⊆ B_{0}. Second we note that by Lemma 5.4, the C^{0,α} estimates for the coefficients of the second order operator Δ_{ϵ,t} depend only polynomially on ϵ^{−1}, and the same is true, much as in the proof of Lemma 5.4, for the constants λ and Λ needed in applying [11], Theorem 6.2. The constant arising in the Schauder estimate can be verified to depend only polynomially on λ and Λ. Taking these things into account, one obtains (5.7).

Now integrating (5.7) with respect to t we obtain

∥u_{ϵ}∥_{C^{2,α}} ≤ C_{3}e^{−C_{4}/ϵ}.

Using Schauder estimates again repeatedly in the standard way (see [35], Formula (4.5) and following text), one can then find for each k, constants C and C′ such that

∥u_{ϵ}∥_{C^{k,α}} ≤ Ce^{−C′/ϵ}.

To get this inequality for any α, one uses the interpolation inequalities. q.e.d.

Remark 5.7. The construction of the Ooguri–Vafa metric in §3 clearly works also for singular fibres of type I_{n}, simply by quotienting at the appropriate stage by ϵnZ instead of ϵZ, and the above proofs go through unchanged in this case. Thus all the results of this section remain valid for elliptic K3 surfaces with semi-stable fibres.

6. Gromov–Hausdorff convergence

We now return to the notion of convergence alluded to in the introduction. We wish to show that with the proper normalization, the results of §5 imply that in the large complex structure limit, K3 surfaces in fact converge to 2-spheres. To make this precise, we first recall the notion of Gromov–Hausdorff distance. The definition given below can be easily seen to be equivalent to a definition in terms of ϵ-dense subsets, c.f. [30] pg. 276.

Definition 6.1. Let (X, d_{X}), (Y, d_{Y}) be two compact metric spaces. Suppose there exists maps f : X → Y and g : Y → X (not necessarily continuous) such that for all x_{1}, x_{2} ∈ X,

|d_{X}(x_{1}, x_{2}) − d_{Y} (f(x_{1}), f(x_{2}))| < ϵ

and for all x ∈ X,

d_{X}(x, g ∘ f(x)) < ϵ,

and the two symmetric properties for Y hold. Then we say the Gromov– Hausdorff distance between X and Y is at most ϵ. The Gromov– Hausdorff distance d_{GH}(X, Y) is the infinum of all such ϵ.

The Gromov–Hausdorff distance defines a topology on the set of compact metric spaces, and hence a notion of convergence. It follows from results of Gromov (see e.g. [30], pg. 281, Cor. 1.11) that the class of compact Ricci-flat manifolds with diameter ≤ D is precompact. Thus in particular, if we have a sequence of Calabi–Yau n-folds whose complex structure converges to a large complex structure limit point (or any other boundary point for that matter) and whose metrics have diameter bounded above, then there is a convergent subsequence, and then the basic question is: what is the limit? The conjecture which motivated the work of this paper is the following:

Conjecture 6.2. Let \overline{M} be a compactified moduli space of complex deformations of a simply-connected Calabi–Yau n-fold X with holonomy group SU(n), and let p ∈ \overline{M} be a large complex structure limit point (see [27] for the precise Hodge-theoretic definition of this notion). Let (X_{i}, g_{i}) be a sequence of Calabi–Yau manifolds with Ricci-flat Kähler metric which are complex deformations of X, with the sequence [X_{i}] ∈ \overline{M} converging suitably to p, and C_{1} ≥ Diam(X_{i}) ≥ C_{2} > 0 for all i. Then a subsequence of (X_{i}, g_{i}) converges to a metric space (X_{∞}, d_{∞}), where X_{∞} is homeomorphic to S^{n}. Furthermore, d_{∞} is induced by a Riemannian metric on X_{∞} \ Δ, with Δ ⊆ X_{∞} a set of codimension 2.

A similar conjecture was also made by Kontsevich, Soibelman and Todorov (see [22], [25]).

Remark 6.3. Conjecture 6.2 is obvious in the elliptic curve case (ignoring the fact that elliptic curves are not simply-connected), no matter how the sequence of points approaches the large complex structure limit point. However, in the K3 case, more care must be taken. In this paper, we have considered limits mirror to points approaching the large Kähler limit along a ray in the Kähler cone. However, if a sequence of points approaching the large Kähler limit approaches the boundary of the projectivized Kähler cone, we might expect further degeneracies in the Gromov-Hausdorff limits. For example, a product of two elliptic curves E_{1} × E_{2} = R^{4}/Z^{4}, with a metric

(\begin{matrix}ϵ^{−3} & 0 & 0 & 0 \\ 0 & ϵ & 0 & 0 \\ 0 & 0 & ϵ & 0 \\ 0 & 0 & 0 & ϵ \end{matrix})

has a special Lagrangian fibration given by projection on the the first and third factors, and has fibres of area ϵ. When we normalise the metrics to have diameter one, the sequence of Riemannian manifolds converges to an S^{1} as ϵ → 0. As pointed out to us by N.C. Leung, this construction descends to the corresponding Kummer surfaces. The limit of the Kummer surfaces is then a closed interval.

Thus we expect that the correct restriction on sequences of points in the complex moduli space in Conjecture 6.2 should correspond in the mirror to Kähler classes staying within a proper subcone of the Kähler cone. We can now prove the conjecture for the limits of K3 surfaces considered in this paper, where the Kähler class tends to ∞ along a ray, which we have seen reduces to the following result.

Theorem 6.4. Let j : J → B be an elliptically fibred K3 surface with a section and singular fibres all of type I_{1}, and let f_{i} : X_{i} → B be a sequence of elliptically fibred K3 surfaces with jacobian j. Let ω_{i} correspond to a Ricci-flat Kähler metric on X_{i} with ω_{i}^{2} independent of i, and with ∫_{f_{i}^{−1}(b)} ω_{i} = ϵ_{i} → 0 as i → ∞. Then the sequence of Riemannian manifolds (X_{i}, ϵ_{i}ω_{i}) converges in the Gromov–Hausdorff sense to B, the metric on B being induced from the (singular) Riemannian metric given, in local coordinates, by W_{0}^{−1}dy ⊗ dȳ, with W_{0} as defined in §4.

Proof. As usual, after choosing a topological zero-section of each X_{i}, we can identify X_{i} with J as a manifold. We may then view the ω_{i} as corresponding to a sequence of Riemannian metrics g_{i} on J, and prove that the sequence J_{i} = (J, ϵ_{i}g_{i}) converges in the Gromov–Hausdorff sense to B (with the given metric).

Using Remark 4.6, we can choose the class B_{i} determining ω_{i} in a fundamental domain for the B-field by making, for each i, a judicious choice of zero-section σ_{0}.

Consider now B along with the metric W_{0}^{−1}dy ⊗ dȳ. Near each singular fibre one can find a coordinate y so that τ_{1} = 1 and τ_{2} = \frac{1}{2πi} log y +h(y), for some holomorphic function h, and from this one can see that each point of Δ ⊆ B is at finite distance under this metric, and thus B becomes a compact metric space using geodesic distance.

Now we need to show that for each δ > 0, d_{GH}(J_{i}, B) < δ for sufficiently large i. We will apply Definition 6.1 to the maps f_{i} = j : J → B and σ_{0} : B → J.

Choose, using Corollary 3.7 and Lemma 5.3, for each point p_{j} ∈ Δ, a small disc D_{j} around p_{j} with the property that:

(1) Diam(D_{j}) < δ/100.

(2) Diam(f^{−1}(D_{j})) < δ/100 for sufficiently small ϵ_{i}.

Let U = B \ ⋃ D_{j}. Now let x_{1}, x_{2} ∈ J. Let γ be a path joining x_{1} and x_{2} such that, for a given i,

l_{ϵ_{i}g_{i}}(γ) < d_{ϵ_{i}g_{i}}(x_{1}, x_{2}) + δ/100.

Here l denotes length, and the subscript denotes the metric being used. At the risk of increasing the length of γ by 24δ/100, we can assume that γ enters and leaves each f^{−1}(D_{j}) at most once, and write γ = γ_{1} + γ_{2}, with γ_{1} ⊆ f^{−1}(U) and γ_{2} ⊆ f^{−1}(⋃ D_{j}), with l_{ϵ_{i}g_{i}}(γ_{2}) ≤ 24δ/100. Now if f^{−1}(U) carried a semi-flat metric, then f^{−1}(U) → U would in fact be a Riemannian submersion, and distances decrease under submersions. On the other hand, if ϵ_{i} is sufficiently small, it follows from Theorem 5.6 that the metric ϵ_{i}g_{i} is close to a semi-flat metric in the C^{0} sense. Thus for sufficiently large i, depending on δ,

l_{B}(f(γ_{1})) ≤ l_{SF} (γ_{1}) ≤ l_{ϵ_{i}g_{i}}(γ_{1}) + C(ϵ_{i}),

where l_{SF} denotes length with respect to the suitably normalized semiflat metric close to ϵ_{i}g_{i}, and C(ϵ_{i}) is a constant depending on ϵ_{i} (and δ) but independent of the path. Furthermore, C(ϵ_{i}) → 0 as ϵ_{i} → 0. Thus, possibly replacing f(γ_{2}) with a shorter path, we see that

d_{B}(f(x_{1}), f(x_{2})) ≤ l_{B}(f(γ_{1})) + 24δ/100
≤ l_{ϵ_{i}g_{i}}(γ_{1}) + C(ϵ_{i}) + 24δ/100.

Thus for sufficiently small ϵ_{i}, we always have

d_{B}(f(x_{1}), f(x_{2})) < d_{ϵ_{i}g_{i}}(x_{1}, x_{2}) + δ.

Next, let y_{1}, y_{2} ∈ B, and let γ be a path joining y_{1} and y_{2} with

l_{B}(γ) < d_{B}(y_{1}, y_{2}) + δ/100.

As before, we can assume that γ enters and leaves each D_{i} once, and write γ = γ_{1} + γ_{2}. Consider now the metric on σ_{0}(B); locally, this takes the form ϵ_{i}(W^{−1} + W |b|^{2})dy ⊗ dȳ for some W and b. Again, the metric on f^{−1}(U) is close to a semi-flat metric, hence this metric is close, in the C^{0} sense, to (W_{0}^{−1} +ϵ_{i}^{2}W_{0}|b_{SF} |^{2})dy ⊗dȳ. Now the point of choosing B_{i} to be in a fundamental domain for the B-field is that |b_{SF} |^{2} can then be uniformly bounded, independent of i. Thus for small ϵ_{i}, this metric is close to the given metric on B. Thus there exists a constant C(ϵ_{i}) with C(ϵ_{i}) → 0 as ϵ_{i} → 0 such that

l_{ϵ_{i}g_{i}}(σ_{0}(γ_{1})) ≤ l_{B}(γ_{1}) + C(ϵ_{i}).

Therefore d_{ϵ_{i}g_{i}}(σ_{0}(y_{1}), σ_{0}(y_{2})) ≤ l_{B}(γ) + C(ϵ_{i}) + 24δ/100 so for sufficiently small ϵ_{i},

d_{ϵ_{i}g_{i}}(σ_{0}(y_{1}), σ_{0}(y_{2})) ≤ d_{B}(y_{1}, y_{2}) + δ.

Thus for sufficiently small ϵ,

|d_{B}(y_{1}, y_{2}) − d_{ϵ_{i}g_{i}}(σ_{0}(y_{1}), σ_{0}(y_{2}))| < δ

for all y_{1}, y_{2} ∈ B.

If x_{1}, x_{2} ∈ J, similar arguments show that

d_{ϵ_{i}g_{i}}(x_{1}, x_{2}) < d_{B}(f(x_{1}), f(x_{2})) + δ

by joining x_{1} and x_{2} by a path which first connects x_{1} to σ_{0}(B) inside a fibre or inside f^{−1}(D_{j}) for some j, then follows a geodesic inside σ_{0}(B) to the fibre containing x_{2}, and then connects up to x_{2} inside this fibre. The inequality follows for sufficiently small ϵ_{i} since the diameter with respect to ϵ_{i}g_{i} of any fibre f^{−1}(y) for y ∈ U, for small ϵ_{i}, is bounded by Cϵ_{i}, where C depends only on the periods over U.

This shows

|d_{ϵ_{i}g_{i}}(x_{1}, x_{2}) − d_{B}(f(x_{1}), f(x_{2}))| < δ

for sufficiently small ϵ_{i}. Finally, similar methods show

|d_{ϵ_{i}g_{i}}(x_{1}, x_{2}) − d_{ϵ_{i}g_{i}}(σ_{0}(f(x_{1})), σ_{0}(f(x_{2}))| < δ

for all x_{1}, x_{2} ∈ X, and ϵ_{i} sufficiently small. q.e.d.

Remark 6.5. The metric on the base B is McLean’s metric (see [26], [19], [14]) on the base of the special Lagrangian T^{2}-fibration obtained by hyperkähler rotation. In higher dimensions we also expect this metric to appear in the limit, showing a residual effect of the conjectural special Lagrangian fibration. This metric would then be singular along some subset of the limit, corresponding to the limit of the discriminant loci of the conjectural special Lagrangian fibrations. We hope this will be codimension 2. See [16] for further speculation along these lines.

Conversely, we hope that one approach to understanding the existence of special Lagrangian fibrations would be to prove Conjecture 6.2, which gives us insight into the behaviour of Ricci-flat metrics near large complex structure limits. However, it is clear that any approach to prove Conjecture 6.2 in higher dimensions must be substantially different to the one given here for K3 surfaces, where we have made use of the existence of special Lagrangian fibrations as well as the hyperkähler trick to reduce to a question of Kähler degenerations.

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University of Warwick, Coventry, U.K. University of Cambridge, Cambridge, U.K.