14 research outputs found
Toric Submanifolds associated to affine subspaces and SYZ Mirror Symmetry
We study Strominger-Yau-Zaslow mirror symmetry for toric manifolds from the
viewpoint of complex submanifolds. Inspired by Yamamoto's work on the semi-flat
case, we develop certain submanifolds in toric manifolds from affine subspaces
in and show that such submanifolds are toric. We further show
that the mirror partner of toric submanifolds is a submanifold with boundary
and corners in a Delzant polytope of the toric manifold.Comment: 48 pages, 20 figure
Singular Lagrangian torus fibrations on the smoothing of algebraic cones
Given a lattice polytope , we can consider the cone
, and the affine toric variety
associated to . Altmann showed that the versal deformation
space of can be described by the Minkowski decomposition of the
polytope . Under some conditions on , we can obtain a smooth deformation
of using Altmann's result. In this article, we
construct a complex fibration on , with general fibre
and finite singular fibres described in terms of the
components of the Minkowski decomposition. We construct a singular Lagrangian
torus fibration out of the complex fibration. This singular fibration admits a
convex base diagram representation with cuts as a natural generalization of
base diagrams described in the work of Symington for Almost Toric Fibrations
(). In particular, we obtain a convex base diagram whose image is the
dual cone of . There is a 1-parameter family of monotone Lagrangian tori
in each of these fibrations. Using the wall-crossing formula, we describe the
potential associated with this family in terms of the Minkowski decomposition
of and discuss non-displaceability. We also discuss some other consequences
of our results.Comment: 43 pages, 14 figure