Some piece-wise smooth Lagrangian fibrations

This paper was motivated by the Strominger-Yau-Zaslow [A. Strominger, S.-T. Yau and E. Zaslow, Nuclear Phys. B 479 (1996), no. 1-2, 243–259; MR1429831 (97j:32022)] conjecture and work surrounding it. This conjecture predicts that mirror symmetry can be explained in terms of dualizing special Lagrangian fibrations on Calabi-Yau manifolds. The paper under review deals with the question of constructing Lagrangian, rather than special Lagrangian, fibrations. One way to construct a smooth Lagrangian torus bundle is to start with an affine manifold, i.e., a real manifold with transition maps in ${ m Aff}({f R}^n)$, whose transition maps in fact have integral linear part. Then there is a local system $Lambda$ contained in the cotangent bundle $T^*B$ of $B$, generated locally by $dy_1,dots,dy_n$ where $y_1,dots,y_n$ are local affine coordinates. Because of the restriction on transition maps, $Lambda$ is well-defined, and $X(B)coloneq T^*B/Lambda$ inherits the canonical symplectic form on $T^*B$ and is a Lagrangian torus bundle over $B$. Now the basic problem is that interesting Lagrangian fibrations will have singular fibres. One considers affine manifolds $B$ with singularities, i.e., topological manifolds $B$ with a dense open set $B_0subseteq B$ which has an affine structure. Ideally, $Deltacoloneq Bsbs B_0$ should have codimension two in $B$. One then seeks compactifications $X(B_0)subset X(B)$ as symplectic manifolds. Of course, one's ability to do this will depend on the nature of the affine structure around $Delta$. It is not difficult to carry this out in two dimensions for some standard types of singularities [see, for example, M. Symington, in Topology and geometry of manifolds (Athens, GA, 2001), 153–208, Proc. Sympos. Pure Math., 71, Amer. Math. Soc., Providence, RI, 2003; MR2024634 (2005b:53142)]. The paper under review is concerned with aspects which only arise in higher dimensions. In particular, it appears that in higher dimensions there are some naturally occurring singularities which can only be compactified using piecewise smooth fibrations. This phenomenon was first seen in work of W.-D. Ruan [in Winter School on Mirror Symmetry, Vector Bundles and Lagrangian Submanifolds (Cambridge, MA, 1999), 297–332, Amer. Math. Soc., Providence, RI, 2001; MR1876075 (2002m:32041)] and was demonstrated by D. D. Joyce [see, for example, Comm. Anal. Geom. 11 (2003), no. 5, 859–907; MR2032503 (2004m:53094)] to be crucial in understanding the Strominger-Yau-Zaslow conjecture. The paper under review is partly expository and partly an introduction to some new ideas of the authors. It begins with a nice exposition of the basic problems and examples that arise in this context, and then proceeds to give some general constructions for producing piecewise smooth Lagrangian fibrations, including ones which have the correct topology for compactifying the "negative vertex'', one of the two basic singularities which occur in three-dimensional Calabi-Yaus. This is the hard case; local models for the "positive vertex'' have been known for a long time. This example has the feature that the discriminant locus is not codimension two, but is a codimension one fattening of a trivalent graph. This appears to be a necessary feature of such examples. The authors then consider periods of such piecewise smooth fibrations, and give some hints at upcoming work on more powerful methods of constructing piecewise smooth fibrations.