Seidel's long exact sequence on Calabi-Yau manifolds

TL;DR

This paper extends Seidel's long exact sequence of Lagrangian Floer cohomology to compact Lagrangian submanifolds with vanishing Maslov class on Calabi-Yau manifolds, using anchored Lagrangian frameworks and compactness results.

Contribution

It generalizes Seidel's sequence to a broader class of Lagrangian submanifolds on Calabi-Yau manifolds employing new compactness theorems for pseudoholomorphic curves.

Findings

Established a generalized long exact sequence for Lagrangian Floer cohomology.

Proved a compactness theorem for smooth J-holomorphic sections in Lefschetz fibrations.

Extended Floer theory techniques to noncompact symplectic manifolds with cylindrical ends.

Abstract

In this paper, we generalize construction of Seidel's long exact sequence of Lagrangian Floer cohomology to that of compact Lagrangian submanifolds with vanishing Malsov class on general Calabi-Yau manifolds. We use the framework of anchored Lagrangian submanifolds developed in \cite{fooo:anchor} and some compactness theorem of \emph{smooth} $J$-holomorphic sections of Lefschetz Hamiltonian fibration for a generic choice of $J$. The proof of the latter compactness theorem involves a study of proper pseudoholomorphic curves in the setting of noncompact symplectic manifolds with cylindrical ends.