ScalingStacks

For general information on this section, see [42, Chapter 8,9], and the introduction in [54]. In the initial years following the SYZ proposal, there was an overly optimistic belief based on the analogy with the hyperkähler case, and based on topological and complex geometric considerations:

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  The special Lagrangian fibration exists globally and is defined by a $C^{\infty}$ map, even though some fibres may be singular.

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  The discriminant locus on the base is codimension two. For Calabi-Yau 3-folds, under suitable genericity assumption, the discriminant locus is a trivalent graph, with two types of vertices, known as positive and negative vertices.³³ 3 The names ‘positive/negative vertex’ come from some old fashioned topological models of the torus fibration where the most singular fibres have Euler characteristics $\pm 1$ respectively.

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  Along the edges of the trivalent graph, the singularity of the SYZ fibration is transversely modelled on the $I_{1}$ singularity.

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  The local region near the positive vertex is complex geometrically a large open subset inside

  $$\{z_{0}z_{1}z_{2}=1-z_{3}\}\subset\mathbb{C}^{3}\times\mathbb{C}^{*}_{z_{3}},\quad\Omega\propto\frac{1}{z_{3}}dz_{0}\wedge dz_{1}\wedge dz_{2},$$

  while the negative vertex region is modelled on a large open subset inside

  $$\{z_{3}z_{4}=1-z_{1}-z_{2}\}\subset(\mathbb{C}^{*})^{2}_{z_{1},z_{2}}\times\mathbb{C}^{2}_{z_{3},z_{4}},\quad\Omega\propto\frac{1}{z_{1}z_{2}}dz_{2}\wedge dz_{3}\wedge dz_{4}.$$

The naïvete was challenged by Joyce [41], based on his observations concerning special Lagrangian singularities, which are markedly different from those visible in holomorphic fibrations. A global special Lagrangian fibration on the compact Calabi-Yau manifolds near the large complex structure, should it exist at all, is expected to have a much more subtle structure:

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  The special Lagrangian fibration is typically not defined by $C^{\infty}$ maps, but are at best piecewise smooth.

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  The discriminant locus on the base is typically not codimension two, but the trivalent graph is expected to be thickened to a codimension one ‘ribbon’. The amount of thickening probably tends to zero in the large complex structure limit.

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  The $I_{1}\times\mathbb{R}$ local singularity model does not lead to a Fredholm deformation theory for the singular special Lagrangian fibres, so should be replaced by some other singularity models.

Stepping aside from the substantial difficulties of the special Lagrangian local singularities, another major difficulty is to understand the Calabi-Yau metrics near the large complex structure limit, in complex dimension three. The optimistic expectations are inspired by the Gross-Wilson picture in complex dimension two. Hypothetically, the Calabi-Yau 3-fold is Gromov-Hausdorff close to a real 3-dimensional manifold, whose topology is believed to be the 3-sphere⁴⁴ 4 This expectation comes from the topology of the essential skeleton, see section 3.3 below., containing a trivialent graph, such that

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  Away from the trivalent graph, the Calabi-Yau metric is semiflat up to exponentially small errors.

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  Transverse to the edges in the trivalent graph, the metric is modelled on the Ooguri-Vafa metric appearing in Gross and Wilson’s picture.

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  Near the positive and negative vertices of the trivalent graph, the local metric is modelled on some generalization of the Ooguri-Vafa metrics.

It was recently realized that there exist almost canonical constructions of Ooguri-Vafa type metrics in complex dimension three, with the predicted topology and complex structure of the positive and negative vertices, constructed from a (nonlinear) generalized Gibbons-Hawking ansatz [54]. The asymptotic geometry of these Ooguri-Vafa type metrics matches with semiflat metrics in the generic region. The positive vertex metric contains a local region modelled on a generalized Taub-NUT type metric on $\mathbb{C}^{3}$, analogous to the way the Ooguri-Vafa metric contains a region modelled on the Taub-NUT metric.

The following questions are widely open: