A new approach to the symplectic isotopy problem

TL;DR

This paper introduces a novel approach to the symplectic isotopy problem in $CP^2$, linking it to the existence of embedded Lagrangian disks and shifting the focus to Lagrangian Floer theory for potential solutions.

Contribution

It establishes an equivalence between the symplectic isotopy problem and the existence of certain Lagrangian disks, offering a new pathway for future research.

Findings

Reduces the symplectic isotopy problem to an Lagrangian disk existence problem.

Provides a new method using nodal symplectic isotopies and model isotopies.

Suggests potential for either proving or disproving the conjecture through this equivalence.

Abstract

The symplectic isotopy conjecture states that every smooth symplectic surface in $CP^2$ is symplectically isotopic to a complex algebraic curve. Progress began with Gromov's pseudoholomorphic curves [Gro85], and progressed further culminating in Siebert and Tian's proof of the conjecture up to degree 17 [ST05], but further progress has stalled. In this article we provide a new direction of attack on this problem. Using a solution to a nodal symplectic isotopy problem we guide model symplectic isotopies of smooth surfaces. This results in an equivalence between the smooth symplectic isotopy problem and an existence problem of certain embedded Lagrangian disks. This redirects study of this problem from the realm of pseudoholomorphic curves of high genus to the realm of Lagrangians and Floer theory. Because the main theorem is an equivalence going both directions, it could theoretically be used to either prove or disprove the symplectic isotopy conjecture.