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 ${\mathbb{C}\mathbb{P}}^5$ and $S^6\times {\mathbb{C}\mathbb{P}}^2$. 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.