Real symplectic formulation of local special geometry

Special Lagrangian Geometry in irreducible symplectic 4-folds

Having fixed a Kaehler class and the unique corresponding hyperkaehler metric, we prove that all special Lagrangian submanifolds of an irreducible symplectic 4-fold X are obtained by complex submanifolds via a generalization of the so called hyperkaehler rotation trick; thus they retain part of the rigidity of the complex submanifolds: indeed all special Lagrangian submanifolds of X turn out to...

Symplectic Geometry

1 Symplectic Manifolds 5 1.1 Symplectic Linear Algebra . . . . . . . . . . . . . . . . . . . . . . . 5 1.2 Symplectic Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.3 Cotangent Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.4 Moser’s Trick . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.5 Darboux and Moser Theorems . . . . . . . . . . . . . . . ....

Symplectic Geometry

Symplectic Geometry

Symplectic Geometry

These notes are written for a ten week graduate class on symplectic geometry. Most of the material here is included in Michèle Audin’s book Torus actions on symplectic manifolds, which I used heavily in preparing these notes. There are a few topics that we cover that are not in Audin’s book, such as polygon spaces, cohomology of flag manifolds, and CS/GKM theory. More significant, however, are ...