We give a method to construct singular Lagrangian 3-torus fibrations over certain a priori given integral affine manifolds with singularities, which we call simple. The main result of this article is the proof that M. Gross' topological Calabi-Yau compactifications [7] can be made into symplectic compactifications. As an example, we obtain a pair of compact symplectic 6-manifolds together with Lagrangian fibrations whose underlying affine structures are dual. The symplectic manifolds obtained are homeomorphic to a smooth quintic Calabi-Yau 3-fold and its mirror.
We present a study on the integral forms and theirČech and de Rham cohomology. We analyze the problem from a general perspective of sheaf theory and we explore examples in superprojective manifolds. Integral forms are fundamental in the theory of integration in supermanifold. One can define the integral forms introducing a new sheaf containing, among other objects, the new basic forms δ(dθ) where the symbol δ has the usual formal properties of Dirac's delta distribution and acts on functions and forms as a Dirac measure. They satisfy in addition some new relations on the sheaf. It turns out that the enlarged sheaf of integral and "ordinary" superforms contains also forms of "negative degree" and, moreover, due to the additional relations introduced it is, in a non trivial way, different from the usual superform cohomology.
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.
We construct a Lagrangian submanifold, inside the cotangent bundle of a real torus, which we call a Lagrangian pair of pants. It is given as the graph of an exact one form on the real blowup of a Lagrangian coamoeba. Lagrangian pairs of pants are the main building blocks in a construction of smooth Lagrangian submanifolds of $( {\mathbb{C}}^*)^n$ that lift tropical subvarieties in $\mathbb R^n$. As an example we explain how to lift tropical curves in $ {\mathbb{R}}^2$ to Lagrangian submanifolds of $( {\mathbb{C}}^*)^2$. We also give several new examples of Lagrangian submanifolds inside toric varieties, some of which are monotone.
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