Circle actions on four dimensional almost complex manifolds with discrete fixed point sets

• Published in: arXiv.org
• Main Author: Jang, Donghoon
• Format: Paper Journal Article
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 08.11.2023
• Subjects: Classification; Estimating techniques; Lower bounds; Mathematics - Differential Geometry; Questions; Rapid prototyping
• ISSN: 2331-8422
• DOI: 10.48550/arxiv.1912.04128

We establish a necessary and sufficient condition for pairs of integers to arise as the weights at the fixed points of an effective circle action on a compact almost complex 4-manifold with a discrete fixed point set. As an application, we provide a necessary and sufficient condition for a pair of integers to arise as the Chern numbers of such an action, answering negatively a question by Sabatini whether \(c_1^2[M] \leq 3 c_2[M]\) holds for any such manifold \(M\). We achieve this by demonstrating that pairs of integers that arise as weights of a circle action, also arise as weights of a restriction of a \(\mathbb{T}^2\)-action. Furthermore, we discuss applications to circle actions on complex/symplectic 4-manifolds and semi-free circle actions with discrete fixed point sets.