The geometry of maximal components of the $\mathsf{PSp}(4,\mathbb R)$ character variety

Abstract

We describe the space of maximal components of the character variety of surface group representations into $\mathsf{PSp}\left(4,ℝ\right)$ and $\mathsf{Sp}\left(4,ℝ\right)$.

For every real rank $2$ Lie group of Hermitian type, we construct a mapping class group invariant complex structure on the maximal components. For the groups $\mathsf{PSp}\left(4,ℝ\right)$ and $\mathsf{Sp}\left(4,ℝ\right)$, we give a mapping class group invariant parametrization of each maximal component as an explicit holomorphic fiber bundle over Teichmüller space. Special attention is put on the connected components which are singular: we give a precise local description of the singularities and their geometric interpretation. We also describe the quotient of the maximal components of $\mathsf{PSp}\left(4,ℝ\right)$ and $\mathsf{Sp}\left(4,ℝ\right)$ by the action of the mapping class group as a holomorphic submersion over the moduli space of curves.

These results are proven in two steps: first we use Higgs bundles to give a nonmapping class group equivariant parametrization, then we prove an analog of Labourie’s conjecture for maximal $\mathsf{PSp}\left(4,ℝ\right)$–representations.