# The symplectic isotopy problem for rational cuspidal curves

## TL;DR

This paper investigates the symplectic isotopy problem for rational cuspidal curves in 4-manifolds, establishing isotopy results up to degree 5 and for certain singularities, using pseudo-holomorphic curves and topology techniques.

## Contribution

It introduces a class of tame singular symplectic curves and proves isotopy to complex curves for degrees up to 5, advancing understanding of symplectic isotopy classes.

## Key findings

- Every rational cuspidal symplectic curve of degree up to 5 is isotopic to a complex curve.
- Curves with one singularity linked to a torus knot are also isotopic to complex curves.
- Classification relies on pseudo-holomorphic curves and symplectic birational geometry techniques.

## Abstract

We define a suitably tame class of singular symplectic curves in 4-manifolds, namely those whose singularities are modeled on complex curve singularities. We study the corresponding symplectic isotopy problem, with a focus on rational curves with irreducible singularities (rational cuspidal curves) in the complex projective plane. We prove that every such curve is isotopic to a complex curve in degrees up to 5, and for curves with one singularity whose link is a torus knot. Classification results of symplectic isotopy classes rely on pseudo-holomorphic curves together with a symplectic version of birational geometry of log pairs and techniques from 4-dimensional topology.