Lefschetz fibrations and exotic symplectic structures on cotangent bundles of spheres; includes Corrigendum

TL;DR

This paper constructs exotic symplectic structures on cotangent bundles of spheres using Lefschetz fibrations, showing that most choices of vanishing cycles produce nonstandard symplectic manifolds, and includes corrections to previous proofs.

Contribution

It introduces a method to generate nonstandard symplectic structures on cotangent bundles via Lefschetz fibrations with mostly fixed vanishing cycles, correcting earlier results.

Findings

Most vanishing cycle choices yield nonstandard symplectic structures.

Standard cotangent bundle structures are precisely characterized.

Corrections made to previous proofs and statements.

Abstract

We construct open symplectic manifolds which are convex at infinity ("Liouville manifolds") and which are diffeomorphic, but not symplectically isomorphic, to cotangent bundles T^*S^{n+1}, for any n+1 \geq 3. These manifolds are constructed as total spaces of Lefschetz fibrations, where the fibre and all but one of the vanishing cycles are fixed. We show that almost any choice of the last vanishing cycle leads to a nonstandard symplectic structure (those choices which yield standard T^*S^{n+1} can be exactly determined). The Corrigendum changes the statement and proof of Lemma 1.1 in the original paper, which corrects our original description of the diffeomorphism type of the manifolds. We also fill a gap in the original proof of Lemma 1.2.