Circle actions on almost complex manifolds with 4 fixed points

• Published in: Mathematische Zeitschrift Vol. 294; no. 1-2; pp. 287 - 319
• Main Author: Jang, Donghoon
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.02.2020; Springer Nature B.V
• Subjects: Mathematics; Mathematics and Statistics; Rapid prototyping; 57M60; Circle action; 37C55; 58C30; 37C25; Weight; Almost complex manifold; Fixed point
• ISSN: 0025-5874; 1432-1823
• DOI: 10.1007/s00209-019-02267-z

Let the circle act on a compact almost complex manifold M . In this paper, we classify the fixed point data of the action if there are 4 fixed points and the dimension of the manifold is at most 6. By the fixed point data we mean a collection of the multisets of the weights at the fixed points. First, if dim M = 2 , then M is a disjoint union of rotations on two 2-spheres. Second, if dim M = 4 , we prove that the action alikes a circle action on a Hirzebruch surface. Finally, if dim M = 6 , we prove that six types occur for the fixed point data; CP 3 type, complex quadric in CP 4 type, Fano threefold type, S 6 ∪ S 6 type, blow up of a fixed point of a rotation on S 6 type, and unknown type that might possibly be realized as a blow up of S 2 inside a manifold like S 6 . When dim M = 6 , we recover the result by Ahara (J Fac Sci Univ Tokyo Sect IA Math 38(1):47–72, 1991 ) in which the fixed point data is determined if furthermore Todd ( M ) = 1 and c 1 3 ( M ) [ M ] ≠ 0 , and the result by Tolman (Trans Am Math Soc 362(8):3963–3996, 2010 ) in which the fixed point data is determined if furthermore the base manifold admits a symplectic structure and the action is Hamiltonian.