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

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.

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Published 1 January 2007