• DOI:10.1007/s00209-019-02267-z
• Corpus ID: 102354818

Circle actions on almost complex manifolds with 4 fixed points

@article{Jang2017CircleAO,
  title={Circle actions on almost complex manifolds with 4 fixed points},
  author={Donghoon Jang},
  journal={Mathematische Zeitschrift},
  year={2017},
  volume={294},
  pages={287-319}
}

• D. Jang
• Published 28 January 2017
• Mathematics
• Mathematische Zeitschrift

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 $$\dim M=2$$ dim M = 2 , then M is a disjoint union of rotations on two 2-spheres. Second, if $$\dim M=4$$ dim M = 4 , we prove that the action alikes a circle action on a Hirzebruch surface. Finally, if $$\dim M=6$$ dim M = 6 , we prove that six types occur for the fixed point data; $$\mathbb {CP}^3$$ CP 3 type, complex quadric in $$\mathbb {CP}^4$$ CP 4 type, Fano threefold type, $$S^6 \cup S^6$$ S 6 ∪ S 6 type, blow up of a fixed point of a rotation on $$S^6$$ S 6 type, and unknown type that might possibly be realized as a blow up of $$S^2$$ S 2 inside a manifold like $$S^6$$ S 6 . When $$\dim M=6$$ 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 $$\mathrm {Todd}(M)=1$$ Todd ( M ) = 1 and $$c_1^3(M)[M] \ne 0$$ 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.