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Circle actions on almost complex manifolds with 4 fixed points

  • Donghoon JangEmail author
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Abstract

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; \(\mathbb {CP}^3\) type, complex quadric in \(\mathbb {CP}^4\) type, Fano threefold type, \(S^6 \cup S^6\) 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 \(\mathrm {Todd}(M)=1\) and \(c_1^3(M)[M] \ne 0\), 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.

Keywords

Almost complex manifold Circle action Fixed point Weight 

Mathematics Subject Classification

58C30 37C25 37C55 57M60 

Notes

References

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsPusan National UniversityPusanSouth Korea

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