DMM Seminar

Speaker: Kaoru Ono (Hokkaido Univ.)
Title: Lagrangian Floer theory on compact toric manifolds (Part 1)
Date (JST): Mon, Oct 24, 2011, 14:00 - 15:00
Place: Seminar Room A
Abstract: I will present joint works with K. Fukaya, Y.-G. Oh and H. Ohta.
In general, there are obstructions to defining Floer cohomology for arbitrary Lagrangian submanifolds.
We developed the framework of Lagrangian Floer theory in terms of filtered A_{\infty}-algebras and introduced the notion of unobstructedness, weakly unobstructedness, the potential function, etc.
In the case of Lagrangian torus fibers of compact toric manifolds, we find that Floer cohomology is well-defined and governed by the potential function.
We can derive some consequences in symplectic geometry such as non-displaceablity and/or lower bound of displacement energy for Lagrangian torus fibers, etc.
In order to understand the critical point theory of the potential function, we constructed an isomorphism between the quantum cohomology and suitably defined Jacobian ring of the potential function. This isomorpihsm also respects pairings: Poincare pairing on the quantum cohomology side and something like residue pairing on the Jacobian ring side.