Embeddings of symplectic balls into the complex projective plane

We investigate spaces of symplectic embeddings of n≤ 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 PGL(3,C) on the configuration space of n ordered points in CP² with the action of the symplectomorphism group Symp(CP²) on the space of n embedded symplectic balls.