A Symplectic Isotopy of a Dehn Twist on CP^n x CP^{n+1}

TL;DR

This paper proves that a specific Dehn twist on the symplectic manifold CP^n x CP^{n+1} is symplectically isotopic to the identity, with an isotopy fixing a hypersurface and lifting to a blow-up, for all ta>1.

Contribution

It demonstrates a symplectic isotopy of a Dehn twist on CP^n x CP^{n+1} that preserves certain submanifolds, extending understanding of symplectic mapping class groups.

Findings

Dehn twist along L^{ta} is isotopic to identity for ta>1

Isotopy fixes a complex hypersurface pointwise

Lifts to the blow-up along a complex submanifold

Abstract

The complex manifold CP^n x CP^{n+1} with symplectic form \sigma_\mu=\sigma_{CP^n}+\mu\sigma_{CP^{n+1}}, where \sigma_{CP^n} and \sigma_{CP^{n+1}} are normalized Fubini-Study forms, n a natural number and \mu>1 a real number, contains a natural Lagrangian sphere L^{\mu}. We prove that the Dehn twist along L^{\mu} is symplectically isotopic to the identity for all \mu>1. This isotopy can be chosen so that it pointwise fixes a complex hypersurface in CP^n x CP^{n+1} and lifts to the blow-up of CP^n x CP^{n+1} along a complex n-dimensional submanifold.