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

TL;DR

The paper addresses the problem of determining the minimal number of fixed points for circle actions on $10$-dimensional compact almost complex manifolds that admit a fixed point. It combines Atiyah–Bott–Berline–Vergne localization, Chern-number relations, and Todd genus calculations to rule out the existence of an action with exactly four fixed points, thereby proving a lower bound of six fixed points; this bound is sharp, realized by $\mathbb{CP}^5$ and $S^6\times\mathbb{CP}^2$. The main technical contributions include vanishing results for certain high-degree Chern numbers under four fixed points and a modular contradiction arising from the GS formula for $\int_M c_1 c_4$. By extending Kosniowski-type bounds to the 10-dimensional setting, the work links local fixed-point data to global topological invariants and confirms sharp, dimension-specific fixed-point behavior for almost complex circle actions.

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.

Theorems & Definitions (18)

• Theorem 1.1
• Theorem 1.2
• Theorem 2.1: ABBV localization theorem
• Theorem 2.2
• Theorem 2.3
• Lemma 2.4
• Theorem 2.5
• Lemma 2.6
• proof
• Theorem 2.7