Circle and torus actions on equal symplectic blow-ups of $\mathbf{\text{CP}^2}$

Abstract

A manifold obtained by $k$ simultaneous symplectic blow-ups of $\CP^2$ of equal sizes $\epsilon$ (where the size of $\CP^1\subset\CP^2$ is one) admits an effective two dimensional torus action if $k \leq 3$ and admits an effective circle action if $\epsilon < 1/(k-1)$. We show that these bounds are sharp if $\epsilon = 1/n$ where $n$ is a natural number. Our proof combines ``soft equivariant techniques with ``hard holomorphic techniques.