Lower bound on the number of fixed points for circle actions on 10-dimensional almost complex manifolds

For a circle action on a compact almost complex manifold with a fixed point, the lower bound on the number of fixed points is known in dimension up to 12 except 10. In this paper, we show that if the circle group acts on a 10-dimensional compact almost complex manifold with a fixed point, then there are at least 6 fixed points. This minimum is attained by $\mathbb{CP}^5$ and $S^6 \times \mathbb{CP}^2$. We establish this lower bound by showing that there does not exist a circle action on a 10-dimensional compact almost complex manifold with 4 fixed points.

💡 Research Summary

The paper addresses a longstanding gap in the classification of circle (S¹) actions on compact almost‑complex manifolds: the minimal number of isolated fixed points in dimension ten. While the lower bound on the number of fixed points is known for dimensions up to twelve, the case of ten dimensions had remained unresolved. The author proves that any effective S¹‑action on a ten‑dimensional compact almost‑complex manifold with at least one fixed point must have at least six fixed points. Moreover, the bound is sharp: the standard linear action on the complex projective space CP⁵ and a diagonal action on the product S⁶ × CP² each realize exactly six fixed points.

The argument proceeds by contradiction. Assuming the existence of a ten‑dimensional almost‑complex manifold M with exactly four isolated fixed points, the author exploits several classical tools:

1. Todd genus and Chern numbers – The Todd genus of a ten‑dimensional almost‑complex manifold can be expressed as
   \