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 $\mathbb{C}\mathrm{P}^2$, normalized so that a line has symplectic area $1$, is homotopy equivalent to the configuration space of $n$ points in $\mathbb{C}\mathrm{P}^2$, provided that the sum of the ball capacities is strictly less than $1$. Our techniques further suggest that, for $n=9$, there are infinitely many homotopy types of spaces of symplectic ball embeddings, depending on the ball capacities. Moreover, for each $n\geq 5$, we exhibit capacities for which the embedding spaces are not simply connected, in contrast with the case $n \leq 4$. As an application, we show that, for $n\geq 9$ equal balls of capacity $c<1/n$, the symplectomorphism group of the blow-up has the homotopy type of the stabilizer of $n$ distinct points in $\mathbb{C}\mathrm{P}^2$.