Embeddings of symplectic balls into the complex projective plane

We investigate spaces of symplectic embeddings of $n\le 4$ balls into the complex projective plane. We prove that they are homotopy equivalent to explicitly described algebraic subspaces of the configuration spaces of $n$ points. We compute the rational homotopy type of these embedding spaces and their cohomology with rational coefficients. Our approach relies on the comparison of the action of $\mathrm{P}\mathrm{G}\mathrm{L}(3,\mathbb{C})$ on the configuration space of $n$ ordered points in ${\mathbf{C}\mathbf{P}}^2$ with the action of the symplectomorphism group $\mathrm{S}\mathrm{y}\mathrm{m}\mathrm{p}({\mathbf{C}\mathbf{P}}^2)$ on the space of $n$ embedded symplectic balls.