Hamiltonian circle action, invariant hypersurface and the complex projective space

TL;DR

This paper characterizes certain symplectic manifolds with Hamiltonian circle actions as homotopy complex projective spaces, under specific invariant hypersurface conditions, extending classical results in transformation group theory.

Contribution

It establishes a new classification result for symplectic manifolds with Hamiltonian circle actions and invariant hypersurfaces, linking them to homotopy complex projective spaces with standard Chern classes.

Findings

$M$ and $D$ are homotopy complex projective spaces with standard Chern classes.

The $S^1$-representations match those from standard linear actions on projective spaces.

The classification holds for $n ot\equiv 3 \pmod 4$.

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.