The symplectic isotopy problem for rational cuspidal curves. Issue 7 (16th July 2022)

• Record Type: Journal Article
• Authors: Golla, Marco; Starkston, Laura
• Is Part Of: Compositio mathematica. Volume 158:Issue 7(2022)
• Page Start: 1595
• Page End: 1682
• Publication Date: 2022-07-16
• Subjects: singular -- symplectic -- isotopy -- rational cuspidal -- plane curve; 57K43 -- 14H50 -- 57R52 -- 14H20; Mathematics -- Periodicals; 510
• DOI: 10.1112/S0010437X2200762X
• Languages: English
• ISSNs: 0010-437X

Abstract

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 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.