Symplectic

Let $\mathrm{Emb}(B^4(c),\mathbb{C}P^2)$ denote the space of unparameterized symplectic embeddings of $k$ balls of capacities $(c_1,\ldots,c_k)$, where $ 1\le k\le 8$. It is known from the work of S. Anjos, J. Li, T.-J. Li, and M. Pinsonnault that the space of capacities decomposes into convex polygons called stability chambers, and that the homotopy type of $\mathrm{Emb}(B^4(c),\mathbb{C}P^2)$ depends solely on the stability chambers. Based on recent results of M. Entov and M. Verbitsky on Kähler-type embeddings, we show that for $1\le k\le 8$, $\mathrm{Emb}(B^4(c),\mathbb{C}P^2)$ is homotopy equivalent to a union of strata $F_I$ of the configuration space of the complex projective plane $F(\mathbb{C}P^2,k)$. The proof relies on constructing an explicit map from the space of Kähler type embeddings to a generalized version of the configuration space that incorporates both configurations of points and compatible complex structures on $\mathbb{C}P^2$.