Exotic symplectic manifolds from Lefschetz fibrations

TL;DR

This paper constructs pairs of Liouville domains in odd complex dimensions that are diffeomorphic but not symplectomorphic, revealing new exotic symplectic structures with distinct Fukaya categories.

Contribution

It introduces explicit examples of exotic symplectic manifolds in all odd complex dimensions using Lefschetz fibrations, demonstrating differences in their Fukaya categories.

Findings

W_0 is symplectomorphic to the cotangent bundle of the sphere.

W_1 has a trivial wrapped Fukaya category.

W_1 contains no compact exact Lagrangian submanifolds.

Abstract

In this paper we construct, in all odd complex dimensions, pairs of Liouville domains W_0 and W_1 which are diffeomorphic to the cotangent bundle of the sphere with one extra subcritical handle, but are not symplectomorphic. While W_0 is symplectically very similar to the cotangent bundle itself, W_1 is more unusual. We use Seidel's exact triangles for Floer cohomology to show that the wrapped Fukaya category of W_1 is trivial. As a corollary we obtain that W_1 contains no compact exact Lagrangian submanifolds.