The symplectic isotopy problem for rational cuspidal curves

2019/07/15 by Golla, Marco, Starkston, Laura · 2 citations
#Algebraic Geometry (math.AG) #FOS: Mathematics #Geometric Topology (math.GT) #Symplectic Geometry (math.SG)

paper · doi:10.48550/arxiv.1907.06787

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.

Cited by

• Surfaces in 4-manifolds and complex curves
• On Weinstein domains in symplectic manifolds

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