Examples of Special Lagrangian Fibrations

Of late there has been a great deal of interest in special Lagrangian submanifolds and manifolds fibred in special Lagrangian submanifolds, motivated by the Strominger-Yau-Zaslow conjecture [29]. One of the basic approaches to finding examples is to exploit symmetries of the ambient manifold. If X is a (non-compact) n-dimensional Calabi-Yau manifold with Kähler form ω and nowhere vanishing holomorphic n-form Ω, and if there is an action of a Lie group G on X preserving these two forms, one can look for G-invariant special Lagrangian submanifolds of X. For us, G will be a torus T m. Using this sort of symmetry to search for examples reduces the special Lagrangian equations to simpler ones which can be solved. This technique has been used independently by M. Haskins and D. Joyce ([12] and [18]) to find new examples of special Lagrangian cones and submanifolds, while it has been used independently by E. Goldstein and myself to construct examples of special Lagrangian fibrations on non-compact Calabi-Yau manifolds with T n−1 actions. This paper is an extended version of an informally distributed preprint (essentially just the first two sections of this paper) released at the same time as Goldstein's preprint [6]. Goldstein has developed his examples in some very interesting directions somewhat orthogonal to the ones taken here. While there is some overlap between the examples considered here (especially in §2) and those in [6], our goal will be to develop more global information about these fibrations. The first set of examples, which we discuss in §2, are special Lagrangian fibrations on crepant resolutions of toric Gorenstein singularities. Such examples were already mentioned in [6]. However, earlier, in [9], I gave topological fibrations on such crepant resolutions , with the belief that these would resemble the actual special Lagrangian fibrations on * Supported in part by the NSF and EPSRC. 1 these manifolds. What is new here is that we show this is indeed the case; with some mild hypotheses on the Kähler metric ω (which we do not require to be Ricci-flat) we find that the topological construction of §3 of [9] coincides with the special Lagrangian fibrations we give here. This gives us a global understanding of the structure of these fibrations, complementing the results in [6]. We also give a new variant of this construction which yields proper fibrations. The other main set of examples, which we consider in §3, …