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

TL;DR

This paper establishes that any circle action on a 10-dimensional compact almost complex manifold with a fixed point must have at least 6 fixed points, filling a gap in the known bounds for such manifolds.

Contribution

The paper proves a new lower bound of 6 fixed points for circle actions on 10-dimensional almost complex manifolds, which was previously unknown.

Findings

Minimum of 6 fixed points for circle actions on 10-dimensional manifolds.

Examples attaining this minimum include $\\mathbb{CP}^5$ and $S^6 \times \mathbb{CP}^2$.

No circle action with only 4 fixed points exists on such manifolds.

Abstract

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.