Circle actions on 8-dimensional almost complex manifolds with 4 fixed points
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Journal of Fixed Point Theory and Applications
Circle actions on 8-dimensional almost complex manifolds with 4 fixed points Donghoon Jang Abstract. In this paper, we prove that if the circle acts on an 8-dimensional compact almost complex manifold M with 4 fixed points, all the Chern numbers and the Hirzebruch χy -genus of M agree with those of S 2 × S 6 . In particular, M is unitary cobordant to S 2 × S 6 . Mathematics Subject Classification. Primary 58C30; Secondary 37C25, 37C55. Keywords. Almost complex manifold, circle action, fixed point, weight.
Contents 1. Introduction 2. Background and preliminaries 3. Proof of Theorem 1.1 4. S 2 × S 6 and S 6 × S 6 , and discussion for higher dimensions References
1. Introduction The purpose of this paper is to discuss the classification of circle actions on almost complex manifolds with discrete fixed point sets and give a classification if the dimension of the manifold is eight and if there are exactly four fixed points. Let the circle act on a 2n-dimensional almost complex manifold (M, J). Throughout this paper, any circle action on an almost complex manifold (M, J) is assumed to preserve the almost complex structure J. Let p be an Donghoon Jang is supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2018R1D1A1B07049511). 0123456789().: V,-vol
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isolated fixed point. Then the local action of the circle near p can be identified with g · (z1 , . . . , zn ) = (g wp,1 z1 , . . . , g wp,n zn ), where g ∈ S 1 ⊂ C, zi are complex numbers, and wp,i are non-zero integers. The non-zero integers wp,1 , . . . , wp,n are called the weights at p. If the fixed point set is discrete, the weights at the fixed points determine the Hirzebruch χy -genus of M (which determines the Todd genus, the signature, and the Euler characteristic of M ), and the Chern numbers of M (which determine the equivariant cobordism class of M ). We discuss the classification of circle actions on compact almost complex manifolds that have few fixed points. For this, let the circle act on a compact almost complex manifold M . First, suppose that there is exactly one fixed point. Then M must be a point. Second, suppose that there are exactly two fixed points. Then either M is the 2-sphere, or dim M = 6 and the weights at the fixed points agree with those of a circle action on S 6 ; see Theorem 2.11. This is proved in [11], whose proof closely follows the proof for a holomorphic vector field on a complex manifold with simple isolated zeroes by Kosniowski [13]. Pelayo and Tolman proved an analogous result for symplectic S 1 -actions on symplectic manifolds [16]. Third, suppose that there are exactly three fixed points. Then the manifold must be four dimensional, and the weights at the fixed points agree with those a standard linear action on CP2 ; see Theorem 2.11. The result is announced in [11] and is proved by carefully adapting the proof for a symplectic circle action on a compact symplectic manifold with 3 fixed po
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