Embeddings of symplectic balls into the complex projective plane

TL;DR

This paper studies the topology of spaces of symplectic embeddings of up to four balls into the complex projective plane, revealing their homotopy types and cohomology structures.

Contribution

It provides explicit descriptions of the homotopy types of these embedding spaces and computes their rational cohomology, connecting symplectic and algebraic geometry.

Findings

Homotopy equivalence to algebraic subspaces of configuration spaces

Explicit computation of rational homotopy types

Determination of rational cohomology of embedding spaces

Abstract

We investigate spaces of symplectic embeddings of $n\leq 4$ 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 $\mathrm{PGL}(3,\mathbb{C})$ on the configuration space of $n$ ordered points in $\mathbf{CP}^2$ with the action of the symplectomorphism group $\mathrm{Symp}(\mathbf{CP}^2)$ on the space of $n$ embedded symplectic balls.