Complicated Singularities – Why Should the Discriminant Locus Change Under Dualizing?

Consider the surface S ⊆ (C^∗)² defined by the equation z₁ + z₂ + 1 = 0. Define the map log : (C^∗)² → R² by log(z₁, z₂) = (log|z₁|, log|z₂|). Then log(S) can be seen as follows. Consider the image of S under the absolute value map.

The line segment r₁ + r₂ = 1 with r₁, r₂ ≥ 0 is the image of {(−a, a−1)|0 < a < 1} ⊆ S; the ray r₂ = r₁ + 1 with r₁ ≥ 0 is the image of {(−a, a−1)|a < 0} ⊆ S; and the ray r₁ = r₂ + 1 is the image of {(−a, a−1)|a > 1} ⊆ S. The map S → |S| is one-to-one on the boundary of |S| and two-to-one in the interior, with (z₁, z₂) and (z̄₁, z̄₂) mapping to the same point in |S|. Taking the logarithm of this picture, we obtain the amoeba of S, log(S) as depicted below.

Now consider S = S × {0} ⊆ Y = (C^∗)² × R = T² × R³. We can now obtain a six-dimensional space X, with a map π : X → Y, an S¹-bundle over Y\S degenerating over S, so that π⁻¹(S) →^≅ S. We then have a T³-fibration on X, f : X → R³, by composing π with the map (log, id) : (C^∗)² × R → R³ = B. Clearly the discriminant locus of f is log(S) × {0}. If b is in the interior of log(S) × {0}, then f⁻¹(b) is obtained topologically by contracting two circles {p₁} × S¹ and {p₂} × S¹ on T³ = T² × S¹ to points. These are the familiar conical singularities seen in the special Lagrangian situation.

If b ∈ ∂(log(S) × {0}), then f⁻¹(b) has a slightly more complicated singularity, but only one. Let us examine how the “generic” singular fiber fits in here. In particular, for b in the interior of log(S) × {0}, locally this discriminant locus splits B into two regions, and these regions represent two different possible smoothings of f⁻¹(b).

Assume now that f : X → B is a special Lagrangian fibration with topology and discriminant locus ∆ being an amoeba. Let b ∈ Int(∆), and set M = f⁻¹(b). Set M^o = M\{x₁, x₂}, where x₁, x₂ are the two conical singularities of M. Suppose that the tangent cones to these two conical singularities, C₁ and C₂, are both cones of the form M⁰. Then the links of these cones, Σ₁ and Σ₂, are T²’s, and one expects that topologically these can be described as follows. Note that M^o ≅ (T²\{y₁, y₂}) × S¹ where y₁, y₂ are two points in T². We assume that the link Σ_i takes the form γ_i × S¹, where γ_i is a simple loop around y_i. If these assumptions hold, then to see how M can be smoothed, we consider the restriction maps in cohomology

H¹(M^o, R) → H¹(Σ₁, R) ⊕ H¹(Σ₂, R)

The image of this map is two-dimensional. Indeed, if we write a basis eⁱ₁, eⁱ₂ of H¹(Σ_i, R) where eⁱ₁ is Poincaré dual to [γ_i] × pt and eⁱ₂ is Poincaré dual to pt × S¹, it is not difficult to see the image of the restriction map is spanned by {(e¹₁, e²₁)} and {(e¹₂, −e²₂)}. Now this model of a topological fibration is not special Lagrangian, so in particular we don’t know exactly how the tangent cones to M at x₁ and x₂ are sitting inside C³, and thus can’t be compared directly with an asymptotically conical smoothing. So to make a plausibility argument, choose new bases fⁱ₁, fⁱ₂ of H¹(Σ_i, R) so that if M(a,0,0), M(0,a,0) and M(0,0,a) are the three possible smoothings of the two singular tangent cones at the singular points x₁, x₂ of M. Then Y(M_i^(a,0,0)) = πafⁱ₁, Y(M_i^(0,a,0)) = πafⁱ₂, and Y(M_i(0,0,a)) = −πa(fⁱ₁ + fⁱ₂).

Suppose that in this new basis, the image of the restriction map is spanned by the pairs (f¹₁, rf²₂) and (rf¹₂, f²₁) for r > 0, r ≠ 1. Then, there are two possible ways of smoothing M, either by gluing in M₁^(a,0,0) and M₂^(0,ra,0) at the singular points x₁ and x₂ respectively, or by gluing in M₁^(0,ra,0) and M₂^(a,0,0) at x₁ and x₂ respectively. This could correspond to deforming M to a fiber over a point on one side of the discriminant locus of f or the other side. This at least gives a plausibility argument for the existence of a special Lagrangian fibration of the topological type given by f. To date, no such fibrations have been constructed, however.

On giving a special Lagrangian fibration with codimension one discriminant and singular fibers with cone over T² singularities, one is just forced to confront a codimension one discriminant locus in special Lagrangian fibrations. This leads inevitably to the conclusion that a “strong form” of the Strominger-Yau-Zaslow conjecture cannot hold. In particular, one is forced to conclude that if f : X → B and f’ : X’ → B are dual special Lagrangian fibrations, then their discriminant loci cannot coincide. Thus one cannot hope for a fiberwise definition of the dualizing process, and one needs to refine the concept of dualizing fibrations. Let us see why the discriminant locus must change under dualizing. The key lies in the behaviour of the positive and negative vertices, where in the positive case the critical locus of the local model of the fibration is a union of three holomorphic curves, while in the negative case the critical locus is a pair of pants. In a “generic” special Lagrangian fibration, we expect the critical locus to remain roughly the same, but its image in the base B will be fattened out. In the negative case, this image will be an amoeba. In the case of the positive vertex, the critical locus, at least locally, consists of a union of three holomorphic curves, so that we expect the discriminant locus to be the union of three different amoebas. The figure below shows the new discriminant locus for these two cases.

Now, under dualizing, positive and negative vertices are interchanged. Thus the discriminant locus must change. This is all quite speculative, of course, and underlying this is the assumption that the discriminant loci are just fattenings of the graphs. However, it is clear that a new notion of dualizing is necessary to cover this eventuality.