All three types of metric ansatzs are constructed in the generalised Gibbons-Hawking framework, by perturbing from the constant solution after incorporating topology. The constant solutions in Example 1.6 serve as zeroth order approximations to the metric ansatz, and can be thought as scaling limits (cf. Section 1.3.3). Geometrically they describe a flat torus fibration fibred over a Euclidean base with distinguished coordinates related to moment maps. The choice of this Euclidean metric is parametrised by a positive definite rank 2 real symmetric or Hermitian matrix, depending on the 3 cases. The principal difference between the Taub-NUT type metric on $\mathbb{C}^{3}$ and the Ooguri-Vafa type metrics is that the bases in the latter cases have periodic directions.

In order to build in the Gross-Ruan-Joyce topology (cf. Section 1.1) we need to make first order corrections to the constant solutions. Recall the generalised Gibbons-Hawking construction involves three sets of equations:

• •

  The integrability condition is responsible for the integrability of the complex structure and the Kähler condition.

• •

  The distributional equation captures the topology and the discriminant locus.

• •

  The Calabi-Yau condition is the only nonlinear equation.

It is natural to impose that the first order corrections satisfy the linearised version of these equations; in particular the linearisation of the Calabi-Yau condition gives rise to harmonic functions. These linearised equations combine into a coupled overdetermined system. Our method to solve this system is to first determine by educated guess the singularities of the harmonic functions along the discriminant locus, explicitly construct such harmonic functions using Green’s representation, and then verify the other equations in the overdetermined system by means of Liouville theorem type arguments. The first order corrections we obtain are canonical (up to constants) under mild growth constraints. For the Taub-NUT type $\mathbb{C}^{3}$ case the first order corrections admit elementary formulae. For the Ooguri-Vafa type metrics on the vertices the first order corrections involve infinite series and Green representation integrals, which are a priori divergent but become convergent after subtracting logarithmically divergent terms, much like what happens already for the Ooguri-Vafa metric.

We then extract various asymptotes of the first order ansatz. Transverse to the discriminant locus, the leading asymptotes can be interpreted geometrically as giving rise to Taub-NUT metrics; ultimately this is forced on us by the distributional equation coming from the topology. In the Ooguri-Vafa type situations, we can also perform Fourier analysis in the periodic variables. Suitably away from the discriminant locus, the zeroth Fourier mode is the dominant contribution, giving rise to a semiflat metric. The harmonicity condition implies that the higher Fourier modes satisfy Helmholtz equations, thereby decay exponentially. An additional problem in the Ooguri-Vafa type situations is that the metric ansatzs are only positive definite on a bounded region, whereby metrically incomplete.

The strategy to identify the holomorphic structure is to produce holomorphic differentials with integral periods, in a manner similar to the Taub-NUT metric and the Ooguri-Vafa metric (cf. Section 1.3.2). The functional equation satisfied by the holomorphic functions allows us to identify the holomorphic volume form. It should be emphasized that while topology is built a priori into the generalised Gibbons-Hawking construction, the holomorphic structure is a nontrivial a posteriori consequence.

The first order corrections are small perturbations suitably away from the discriminant locus, but near the discriminant locus they are large compared to the constant solution. This explains why the first order metric ansatz is approximately Calabi-Yau suitably away from the discriminant locus. In the suitable weighted Hölder norms this approximation continues to hold good near the discriminant locus, except on small balls near the origin in the Taub-NUT type $\mathbb{C}^{3}$ case and the positive vertex case. Geometrically this problem is caused by the 3 edges of $\mathfrak{D}$ interacting strongly at their intersection point. The same problem does not appear on the negative vertex because the discriminant locus $S$ has no singular point.

Our strategy trifurcates at this point. The small ball is a fully nonlinear region in which linear approximation methods fail completely. In the case of the Taub-NUT type metric on $\mathbb{C}^{3}$, we instead shift to the complex geometric perspective, and solve the complex Monge-Ampère equation with prescribed asymptotes at infinity. This is viable because the exterior of the small ball does admit an approximately Calabi-Yau ansatz. The output is a Calabi-Yau metric on $\mathbb{C}^{3}$ whose deviation from the first order ansatz satisfies an asymptotically good estimate.

The Ooguri-Vafa type metric on the positive vertex is best thought as the periodic version of the Taub-NUT type metric on $\mathbb{C}^{3}$, and is obtained by gluing the Taub-NUT type $\mathbb{C}^{3}$ to the first order ansatz on the positive vertex. The periodicity condition breaks down the scaling symmetry of the Taub-NUT type metrics, and instead results in the gluing picture, exactly analogous to the relation between the Taub-NUT metric and the usual Ooguri-Vafa metric. The nonlinear effect on the positive vertex is already fully present on the Taub-NUT type $\mathbb{C}^{3}$. It is worth comparing with the topological prediction of Gross-Ruan (cf. Section 1.1.3) where the neighbourhood of the origin is modelled on $\mathbb{C}^{3}$ with a $T^{2}$ fibration related to the Harvey-Lawson example 1.7. But for metric purposes we need to use an exotic Calabi-Yau metric on $\mathbb{C}^{3}$, rather than the Euclidean $\mathbb{C}^{3}$.

The Ooguri-Vafa type metric on the negative vertex, on the other hand, is constructed entirely perturbatively from the first order ansatz.

The analytic step is aimed at perturbing the first order ansatz into a genuine Calabi-Yau metric, and the techniques involved overlap substantially in all three cases.

A central issue, roughly put, is to produce a parametrix for the right inverse to the Laplacian with accurate control on weighted Hölder norm estimates. Some of the main difficulties are:

• •

  The first order corrected metric is multiscaled, namely it has very different characteristic behaviours in different regions and at different length scales.

• •

  The initial error decays slowly.

The core idea in our methodology is divide and conquer. We decompose the source function according to its support. The contribution supported sufficiently away from the discriminant locus is inverted approximately using the Euclidean Green operator, reflecting the fact that the constant solution is the zeroth order approximation to the metric ansatz. Afterwards the source function is effectively supported near the discriminant locus. We then use a Green operator adapted to the Taub-NUT fibration near the discriminant locus to cure the remaining source.

We now turn to specifics. The 3 cases are arranged in pedagogical order, and each case contains most difficulties of previous cases. As a general policy, detailed proofs will be omitted if the main techniques appeared previously.

In the Taub-NUT type $\mathbb{C}^{3}$ case, the parametrix is used to improve the approximation to the Calabi-Yau condition asymptotically outside a compact region. Once the decay of the approximation error is sufficiently fast, we can appeal to a non-compact version of Yau’s solution to the Calabi conjecture, developed in H-J. Hein’s thesis [12], to turn the ansatz into a genuine Calabi-Yau metric with effective estimates.

Here a difficulty caused by the slow decay of error is that the inverse of the Laplacian is not well behaved in the weighted Hölder spaces. Instead it is preferable to work with the zeroth order operator $\nabla^{2}\Delta^{-1}$, which controls how to correct a Kähler metric for a given amount of volume form error. The advantage is that this operator maps between function spaces with the same Hölder weights, the operator norm is not affected by rescaling the metric, and crucially the Schwartz kernel has two extra order of decay compared to $\Delta^{-1}$.

In the positive vertex case, the main new difficulty is to prove exponential decay of higher Fourier modes. This comes down to mapping properties of the periodic Euclidean Green operator, ultimately thanks to the exponential decay of the higher Fourier modes of the periodic Newtonian potential.

The second new difficulty is that that the volume form error does not decay, and in fact grows logarithmically at large distance, causing problem for perturbation theory over an exponentially long region. The strategy is to first correct the error inside the generic region in the generalised Gibbons-Hawking framework, using the periodic Green operator. We then switch to the complex geometric viewpoint and solve the complex Monge-Ampère equation perturbatively, which avoids the difficulty of the generalised Gibbons-Hawking equation near the discriminant locus.

The third new difficulty comes from metric incompleteness: the Laplacian has no good mapping property in the naïve weighted Hölder spaces. In our approach, this means the parametrix is only defined on compactly supported sources, but the outputs are generally not compactly supported. A formal trick called extension norms [27] effectively allows us to assume the source is compactly supported. This circumvents the need to impose a non-canonical boundary condition.

In the negative vertex case, the main new difficulty comes from the curved nature of the discriminant locus $S$, making it harder to produce a parametrix near $S$. A closely related issue is that there is no obvious a priori choice of smooth topology such that the first order metric ansatz is smooth along $S$. These problems force us to work in weighted Hölder spaces with low regularity, in which it makes no sense to speak of an arbitrarily high order of differentiability. Crucially there is enough regularity to make the Laplacian well defined. The smooth topology emerges a posteriori only after solving the complex Monge-Ampère equation. The solution itself defines a complex structure, hence induces a smooth topology, and the compatibility of the metric with this smooth topology is a consequence of the well known regularity theory for complex Monge-Ampère equation.

We now explain how this paper fits into a program to prove the metric version of the SYZ conjecture for Calabi-Yau 3-folds (cf. Conjecture 1.1). This program runs as follows:

1. (1)

   Produce the metric models on the positive and negative vertices.

2. (2)

   The metric structure near the edges in the Gross-Ruan picture are expected to be modelled on a fibration by Ooguri-Vafa metrics. The problem is that Ooguri-Vafa metrics transverse to the edge depend on a moduli parameter which can vary along the edge, possibly governed by an adiabatic equation.

3. (3)

   The SYZ base $B$ as an affine manifold with singularity along a trivalent graph, can be produced from algebraic geometry in some degree of generality [32][16]. The central problem is then to solve the real Monge-Ampère equation with some prescribed singularities along the trivalent graph. This would allow us to produce a semiflat metric which models the generic region of the SYZ fibration.

4. (4)

   One then glues together the metric models in various regions to obtain the global Calabi-Yau metric on the Calabi-Yau 3-fold, similar to Gross and Wilson’s work on K3 surfaces [11]. Some Fourier analysis is needed to prove exponential decay estimates for deviation from the semiflat metric.

5. (5)

   The existence of the SYZ fibration in the generic region is expected to be a straightforward consequence of the gluing construction. To produce the SYZ fibration near the trivalent graph, one needs to produce models for singular SYZ fibrations on the metric models, and set up a Fredholm deformation theory to ensure the SYZ fibration persists when the metric deforms.

The principal contribution of this paper is to carry out Step (1), and our linear analysis is likely to be useful in Step (4). Some informal digressions in this paper go some way towards addressing difficulties in the other Steps:

In Step (3), the singularity of the real Monge-Ampère equation near the trivalent graph in $B$ should match up with the asymptotic behaviour of the metric models around the trivalent graph, in order to enable the gluing construction in Step (4). This requires understanding how the Ooguri-Vafa type metrics on the vertices transition into the generic region of the SYZ fibration. We propose a mechanism called running coupling for this transition to take place over an exponentially long neck region (cf. Section 3.10 and 4.13). Starting from the observation that Ooguri-Vafa type metrics naturally arise in a family parametrised by some positive definite rank 2 matrices referred to as coupling constants, we argue semi-heuristically that these coupling constants drift slowly as the logarithmic scale increases, governed by an ODE called the renormalistion flow equation which can be solved exactly.

The behaviour of the special Lagrangian fibrations is discussed in Corollary 2.29, Corollary 3.35 and Section 4.12. In both the Taub-NUT type $\mathbb{C}^{3}$ case and the positive vertex case, the $T^{2}$-symmetry provides two symplectic moment coordinates and another real coordinate $\text{Im}(\eta)$, which define a map to $\mathbb{R}^{3}$ whose fibres are $T^{2}$-invariant special Lagrangians with phase zero. However, Joyce’s critique suggests the singularity structure of this SYZ fibration is not stable under metric perturbation.

In the negative vertex case (cf. Section 4.12), there is a homological constraint for the SYZ fibration to exist, namely the Hermitian matrix $a_{p\bar{q}}$ needs to be symmetric. When this constraint holds, we outline a speculative description of a $U(1)$-invariant SYZ fibration on the model metric, and explain how it fits with Joyce’s work on $U(1)$-invariant special Lagrangians. The case where this constraint does not hold is possibly relevant for metric degenerations outside the scope of the SYZ conjecture.