The symplectic isotopy problem for rational cuspidal curves

Compositio mathematica

article

Marco Golla¹  Laura Starkston² 

[1] CNRS, Laboratoire de Mathématiques Jean Leray, Université de Nantes;Department of Mathematics

关键词: singular;    symplectic;    isotopy;    rational cuspidal;    plane curve;    57K43;    14H50;    57R52;    14H20;   

DOI  :  10.1112/S0010437X2200762X

学科分类：数学（综合）

来源: Cambridge University Press

【 摘 要 】

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 five, 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 four-dimensional topology.

【 授权许可】

CC BY