The geometry of maximal components of the PSp(4;ℝ) character variety

We describe the space of maximal components of the character variety of surface group representations into PSp(4;ℝ) and Sp(4;ℝ). 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 PSp(4;ℝ) and Sp(4;ℝ), 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 PSp(4;ℝ) and Sp(4;ℝ) 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 PSp(4;ℝ)–representations.