Embedding more than 8 symplectic balls in $\mathbb{C}\mathrm{P}^2$

Abstract

We prove that the space of symplectic embeddings of $n\geq 1$ standard balls into the standard complex projective plane $\mathbb{C}\mathrm{P}^2$ is homotopy equivalent to the configuration space of $n$ points in $\mathbb{C}\mathrm{P}^2$, provided that the sum of the capacities of the balls is strictly less than the symplectic area of a line. Moreover, our techniques suggest that, for $n=9$, there are infinitely many homotopy types of spaces of symplectic ball embeddings, depending on the capacities of the balls.