Hamiltonian circle action, invariant hypersurface and the complex projective space

TL;DR

The paper proves that a Hamiltonian $S^1$-action on a closed symplectic manifold $M^{2n}$ with isolated fixed points becomes rigid, in the presence of an $S^1$-invariant hypersurface $D$ with $M\setminus D$ a homology cell, forcing $M$ and $D$ to be homotopy complex projective spaces with standard Chern classes. The method blends circle-action techniques (weights at fixed points, Bott residue formula, Hirzebruch $\chi_y$-genus) with Hattori’s analogue of Kobayashi–Ochiai, showing that the fixed-point data mirror the linear action on $(\mathbb{P}^n,\mathbb{P}^{n-1})$ when $n \not\equiv 3 \,\text{mod }4$, and under a Chern-pair caveat otherwise. A key part is deriving that $M$ and $D$ are cohomology (hence homotopy) projective spaces, and that $c_1(M)$ and $c_1(D)$ assume the standard projective values, with the possibility of a special case only when $n \equiv 3 \pmod 4$. The result provides a circle-action analogue to a conjecture of Fujita and its refined versions, linking transformation groups to classical projective-model data. The approach yields explicit descriptions of the $S^1$-weights at fixed points, the Chern numbers, and the topological type, establishing a robust rigidity phenomenon for Hamiltonian circle actions under homology-cell complement hypersurface assumptions.

Abstract

Let $M$ be a $2n$-dimensional closed symplectic manifold admitting a Hamiltonian circle action with isolated fixed points. We show that if $M$ contains an $S^1$-invariant symplectic hypersurface $D$ such that $M\setminus D$ is a homology cell, which is satisfied when $M\setminus D$ is contractible, then $M$ and $D$ are homotopy complex projective spaces with standard Chern classes and the $S^1$-representations on the fixed-point set of $(M,D)$ are the same as those arising from the standard linear actions on $(\mathbb{P}^n,\mathbb{P}^{n-1})$, provided that $n \not \equiv 3 \pmod 4$. This can be viewed as the transformation group analogue to a recent result obtained by Peternell and the author, where the latter was conjectured by Fujita more than four decades ago.

Theorems & Definitions (40)

• Definition 1.1
• Remark 1.2
• Theorem 1.3
• Remark 1.4
• Definition 1.5
• Corollary 1.6
• Lemma 2.1
• Remark 2.2
• Definition 2.3
• Lemma 2.4