Rigidity of Lagrangian submanifolds

Abstract

We study the symplectic geometry and topology related to Lagrangian submanifolds. More specifically we study the following two problems. How do iterated Dehn twists affect the symplectic and Lagrangian topology (in particular Floer homology groups) and how does the existence of certain Lagrangian submanifolds depend on the symplectic structure? First, we review the definition of symplectic Dehn twists and Lagrangian and fixed point Floer homologies. We then construct a chain complex that fits in long exact sequences generalizing Seidel’s exact sequences for a single Dehn twist. This chain complex can be viewed as the difference between Floer homologies before and after the iterated Dehn twists. Next, we study the second problem in the case of Lagrangian projective planes in rational symplectic 4-manifolds. Using previous results specific to dimension 4, we can reduce the problem to describing a finite-dimensional cone in H2(M,R) using finitely many inequalities. Finally, using rational blow-up we can relate Lagrangian projective planes in M with certain symplectic spheres in the blow-up of M, whose dependence on the symplectic structure is known when the blow-up is small. Finally, we use almost toric fibrations to prove the existence of Lagrangian projective planes for many symplectic structures in arbitrarily large blow-ups of the complex projective plane.