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In this article we construct Lagrangian torus fibrations for general quintic Calabi-Yau hypersurfaces near the large complex limit and their mirror manifolds using gradient flow method. Then we prove the Strominger-Yau-Zaslow mirror conjecture for this class of Calabi-Yau manifolds in symplectic category.
We prove a Kawamata-Viehweg vanishing theorem on a normal compact Kähler space X: if L is a nef line bundle with L2 ≠ 0, then H>q(X,KX+L) = 0 for q ≥ dim X − 1. As an application we complete a part of the abundance theorem for minimal Kähler threefolds: if X is a minimal Kähler threefold, then the Kodaira dimension κ(X) is nonnegative.
This is the last in a series of five papers studying compact special Lagrangian submanifolds (SL m-folds) X in (almost) Calabi-Yau m-folds M with singularitiesx1, . . . , xn locally modelled on special Lagrangian cones C1, . . . , Cn in ℂm with isolated singularities at 0. Readers are advised to begin with this paper.
We survey the major results of the previous four papers, giving brief explanations of the proofs. We apply the results to describe the boundary of a moduli space of compact, nonsingular SL m-folds N in M. We prove the existence of special Lagrangian connected sums N1#...#Nk of SL m-folds N1, . . . , Nk in M. We also study SL 3-folds with T2-cone singularities, proving results related to ideas of the author on invariants of Calabi-Yau 3-folds, and the SYZ Conjecture.
Let X be a compact SL m-fold with isolated conical singularities xi and cones Ci for i = 1, . . . , n. The first paper studied the regularity of X near its singular points, and the the second the moduli space of deformations of X. The third and fourth papers construct desingularizations of X, realizing X as a limit of a family of compact, nonsingular SL m-folds Nt in M for small t > 0. Let Li be an asymptotically conical SL m-fold in ℂm asymptotic to Ci at infinity. We make Nt by gluing tLi into X at xi for i = 1, . . . , n.