Generalized special Lagrangian torus fibration for Calabi-Yau hypersurfaces in toric varieties II

In this paper we construct monodromy representing generalized special Lagrangian torus fibrations for Calabi-Yau hypersurfaces in toric varieties near the large complex limit.Comment: A mistake in the proof of the previous lemma 3.4 is corrected. Some arguments and results are simplified and clarifie

Degeneration of Kahler-Einstein hypersurfaces in complex torus to generalized pair of pants decomposition

In this paper we show that the convergence of complete Kahler-Einstein hypersurfaces in complex torus in the sense of Cheeger-Gromov will canonically degenerate the underlying manifolds into "pair of pants" decomposition. We also construct minimal Lagrangian tori that represent the vanishing cycles of the degeneration.Comment: Revised version. Some clarification made and an example adde

Lagrangian torus fibration of quintic Calabi-Yau hypersurfaces I: Fermat quintic case

In this paper we give a construction of Lagrangian torus fibration for Fermat type quintic \cy hypersurfaces via the method of gradient flow. We also compute the monodromy of the expected special Lagrangian torus fibration and discuss structures of singular fibers.Comment: 43 pages, 8 figures. Updated version. Appeared in "Winter School on Mirror Symmetry, Vector Bundles and Lagrangian Submanifolds", edited by S.-T. Yau and C. Vafa, AMS and International Pres

The Fukaya category of symplectic neighborhood of a non-Hausdorff manifold

In this paper, using similar idea as in Fukaya-Oh's work ([9]), we devise a method to compute the Fukaya category of certain exact symplectic manifolds by reducing it to the corresponding Morse category of non-Hausdorff manifold as perturbation of the Lagrangian skeleton of the exact symplectic manifold.Comment: 59 pages, 2 figures. Figures look better in ps fil

H-minimal Lagrangian fibrations in Kahler manifolds and minimal Lagrangian vanishing tori in Kahler-Einstein manifolds

H-minimal Lagrangian submanifolds in general K\"{a}hler manifolds generalize special Lagrangian submanifolds in Calabi-Yau manifolds. In this paper we will use the deformation theory of H-minimal Lagrangian submanifolds in K\"{a}hler manifolds to construct minimal Lagrangian torus in certain K\"{a}hler-Einstein manifolds with negative first Chern class.Comment: 13 pages. We find that the "harmonic Lagrangian" we introduced is equivalent to "H-minimal Lagangian" studied by Oh 10 years ago. Revisions are made accordingly. Some results are simplified and rearrange

Canonical coordinates and Bergman metrics

In this paper we will discuss local coordinates canonically corresponding to a Kahler metric. We will also discuss and prove the $C^\infty$ convergence of Bergman metrics following Tian's result on $C^2$ convergence of Bergman metrics. At the end, we present an interesting characterization of ample line bundle that could be useful in arithmetic geometry.Comment: 44 pages, LaTe

Exponential sums, peak sections, and an alternative version of Donaldson's theorems

In this paper, we provide an alternative proof of Donaldson's almost-holomorphic section theorem and symplectic Lefschetz pencil theorem, through constructions of certain special kind of Donaldson-type sections of the line bundle based on properties of exponential sums.Comment: 28 pages. 2 figures. Figures look better in ps forma

Degeneration of K\"{a}hler-Einstein Manifolds I: The Normal Crossing Case

In this paper we prove that the K\"{a}hler-Einstein metrics for a degeneration family of K\"{a}hler manifolds with ample canonical bundles Gromov-Hausdorff converge to the complete K\"{a}hler-Einstein metric on the smooth part of the central fiber when the central fiber has only normal crossing singularities inside smooth total space. We also prove the incompleteness of the Weil-Peterson metric in this case.Comment: minor correction in referencin

Degeneration of K\"{a}hler-Einstein Manifolds II: The Toroidal Case

In this paper we prove that the K\"{a}hler-Einstein metrics for a toroidal canonical degeneration family of K\"{a}hler manifolds with ample canonical bundles Gromov-Hausdorff converge to the complete K\"{a}hler-Einstein metric on the smooth part of the central fiber when the base locus of the degeneration family is empty. We also prove the incompleteness of the Weil-Peterson metric in this case.Comment: The assumption of simple in the toroidal degeneration is removed using base extensio

Newton polygon and string diagram

In this work, we discuss graph like image of curves under moment maps and their relation with the Newton polygon of the curve, which has applications to Lagrangian torus fibration of Calabi-Yau manifolds.Comment: Revised and update