Special Lagrangian submanifolds of log Calabi–Yau manifolds

Lectures on Special Lagrangian Submanifolds

These notes consist of a study of special Lagrangian submanifolds of Calabi-Yau manifolds and their moduli spaces. The particular case of three dimensions, important in string theory, allows us to introduce the notion of gerbes. These offer an appropriate language for describing many significant features of the Strominger-Yau-Zaslow approach to mirror symmetry.

Lagrangian Submanifolds in Hyperkähler Manifolds, Legendre Transformation

We develop the foundation of the complex symplectic geometry of Lagrangian subvarieties in a hyperkähler manifold. We establish a characterization, a Chern number inequality, topological and geometrical properties of Lagrangian submanifolds. We discuss a category of Lagrangian subvarieties and its relationship with the theory of Lagrangian intersection. We also introduce and study extensively a...

Lagrangian Submanifolds in Hyperkähler manifolds, Legendre transformation

Connected Sums of Special Lagrangian Submanifolds

Let M1 and M2 be special Lagrangian submanifolds of a compact Calabi-Yau manifold X that intersect transversely at a single point. We can then think of M1∪M2 as a singular special Lagrangian submanifold of X with a single isolated singularity. We investigate when we can regularize M1 ∪ M2 in the following sense: There exists a family of Calabi-Yau structures Xα on X and a family of special Lagr...

Deformations of Asymptotically Conical Special Lagrangian Submanifolds

The naive approach is to parametrize these deformations as the zero-set of a “mean curvature operator”, then study them using the implicit function theorem. However, this entails a good understanding of the Jacobi operator of the initial submanifold Σ, which in general is not possible. The work of Oh and, more recently, of McLean (cfr. [Oh], [ML]) shows that, in the “right” geometric context, t...