Related papers: Special Lagrangian submanifolds with isolated coni…
In this paper we construct and classify Lagrangian T^3-fibrations on non compact symplectic manifolds with singular fibres of prescribed topological type. This contributes to the understanding of the structure of the singular fibres that…
This paper explores the topology of monotone Lagrangian submanifolds $L$ inside a symplectic manifold $M$ by exploiting the relationships between the quantum homology of $M$ and various quantum structures associated to the Lagrangian $L$.
Given a lattice polytope $Q\subset \mathbb{R}^n$, we can consider the cone $\sigma=C(Q)=\{\lambda(q,1)\in \mathbb{R}^{n+1}|\lambda \in \mathbb{R}_{\geq0}, q\in Q\} \subset \mathbb{R}^{n+1}$, and the affine toric variety $Y_{\sigma}$…
For every odd natural number g=2d+1 we prove the existence of a countably infinite family of special Lagrangian cones in C^3 over a closed Riemann surface of genus g, using a geometric PDE gluing method.
Locally stable minimal hypersurface could have singularities in dimension $\geq 7$ in general, locally modeled on stable and area-minimizing cones in the Euclidean spaces. In this paper, we present different aspects of how these…
I point out that (BPS saturated) A-type D-branes in superstring compactifications on Calabi-Yau threefolds correspond to {\em graded} special Lagrangian submanifolds, a particular case of the graded Lagrangian submanifolds considered by M.…
In this paper we are interested in quotients of Calabi-Yau threefolds with isolated singularities. In particular, we analyze the case when $X/G$ has terminal singularities. We prove that, if $G$ is cyclic of prime order and $X/G$ has…
In this paper, we show that the calibrated method can also be used to detect indefinite minimal Lagrangian submanifolds in $C_k^m$. We introduce the notion of indefinite special Lagrangian submanifolds in $C_k^m$ and generalize the…
In this article we prove a rigidity theorem for lagrangian singularities by studying the local cohomology of the lagrangian de Rham complex that was introduced in math.AG/0002083. The result can be applied to show the rigidity of all open…
In this paper, we prove a transversality theorem for the moduli space of perturbed special Lagrangian submanifolds in a 6-dimensional manifold equipped with a generalization of a Calabi-Yau structure. These perturbed special Lagrangian…
We characterize the geometry produced by M5-branes wrapping a Special Lagrangian 3-cycle in a Calabi-Yau 3-fold. The presence of the brane replaces the the Calabi-Yau by a real manifold with an almost complex structure. We show that, in…
Let $X$ be a non-singular compact K\"ahler manifold, endowed with an effective divisor $D= \sum (1-\beta_k) Y_k$ having simple normal crossing support, and satisfying $\beta_k \in (0,1)$. The natural objects one has to consider in order to…
This is the third and last in a series of papers working towards the construction of non-trivial Cayley fibrations using gluing methods. In this paper we will show two stability results for Cayley fibrations with certains types of conical…
We obtain new restrictions on Maslov classes of monotone Lagrangian submanifolds of $\mathbb{C}^n$. We also construct families of new examples of monotone Lagrangian submanifolds, which show that the restrictions on Maslov classes are sharp…
We study M-theory fivebranes wrapped on Special Lagrangian submanifolds ($\S_n$) in Calabi-Yau three- and fourfolds. When the M5 wraps a four-cycle, the resulting theory is a two-dimensional domain wall embedded in three-dimensional bulk…
We define the notion of special Lagrangian curvature, showing how it may be interpreted as an alternative higher dimensional generalisation of two dimensional Gaussian curvature. We obtain first a local rigidity result for this curvature…
The branched deformations of immersed compact special Lagrangian submanifolds are studied in this paper. If there exists a nondegenerate $\mathbb{Z}_2$ harmonic 1-form over a special Lagrangian submanifold $L$, we construct a family of…
In this paper we introduce the area of 2-ruled 4-folds in R^n (n=7 or 8), that is, submanifolds M of R^n that admit a fibration over some 2-fold Sigma such that each fibre is an affine 2-plane in R^n. This is motivated by the paper…
The principal theory of this paper comprises a technique for constructing associative, coassociative and Cayley submanifolds of Euclidean space with symmetries, using first-order ordinary differential equations. Explicit examples of…
Let X/G be a 3-dimensional Calabi-Yau orbifold with codimension 2 singularities. The topology of crepant resolutions of X/G is described by the McKay correspondence (Reid, Ito). We study Calabi-Yau 3-folds Y that arise by deforming the…