Semi-Global Invariants of Piecewise Smooth Lagrangian Fibrations

Abstract

We study certain types of piecewise smooth Lagrangian fibrations of smooth symplectic manifolds, which we call stitched Lagrangian fibrations. We extend the classical theory of action-angle coordinates to these fibrations by encoding the information on the non-smoothness into certain invariants consisting, roughly, of a sequence of closed 1-forms on a torus. The main motivation for this work is given by the piecewise smooth Lagrangian fibrations previously constructed by the authors [3], which topologically coincide with the local models used by Gross in Topological Mirror Symmetry [5].the term 'gluing' usually has a smoothness meaning attached to it.

Lagrangian fibrations on blowups of toric varieties and mirror symmetry for hypersurfaces

“…Note that this holomorphic volume form scales with ǫ. This fibration is analogous to some of the examples considered in [23,24,10,11]; see also Example 3.3.1 in [7].…”

“…In Section 4 we construct a Lagrangian torus fibration on X 0 , similar to those previously considered by Gross [23,24] and by Castaño-Bernard and Matessi [10,11]. In Section 5 we study the Lagrangian Floer theory of the torus fibers, which we use to prove Theorem 1.7.…”

Symmetries of Lagrangian fibrations

“…To define a symplectic structure on X, in other words, to achieve a symplectic compactification of X 0 , one needs local models of Lagrangian fibrations with singular fibres. These models were studied in [5], [7], [9]. In dimension n = 2, ∆ consists of a finite collection of points and the symplectic compactification of X 0 is achieved by gluing a standard model of a Lagrangian fibration over a disc with a nodal central fibre; this model is known in symplectic geometry as a simple focus-focus fibration.…”

“…Clearly ∆ has the shape of an amoeba: For a discussion of the topology of the fibres in this example we refer to [7]. Observe that Φ commutes with the conjugation map ι, and therefore ι is fibre-preserving.…”

“…Involutions of stitched fibrations. This Section is rather technical, closely related to [9], and may be skipped on a first reading. The main (new) results are Theorems 3.24 and 3.26 where we provide conditions for the existence of a smooth, fibre-preserving, anti-symplectic involution of stitched fibrations, which will be used to prove the existence of involutions of the negative fibration.…”