Hamiltonian circle action, invariant hypersurface and the complex projective space

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.