Holomorphic curves in Lagrangian torus fibrations.

摘要

This thesis deals with the problem of finding holomorphic curves in symplectic manifolds which are Lagrangian torus fibrations with a well defined action of the torus. A family of complex structures are defined which can be viewed as collapsing the torus fibers. Under this degeneration, it is seen that holomorphic curves converge to objects called holomorphic graphs, similar to what are called tropical curves in the algebraic setting of tropical geometry.; A moduli space of objects called Jepsilon holomorphic graphs is defined, and proved to be cobordant to the moduli space of holomorphic curves. Thus the moduli space of J epsilon holomorphic graphs can be used to calculate invariants of the moduli space of holomorphic curves.