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

Alessandrini, Daniele¹ ; Collier, Brian²

¹ Mathematisches Institut, Universitaet Heidelberg, Heidelberg, Germany
² Department of Mathematics, University of Maryland, College Park, MD, United States

Geometry & topology, Tome 23 (2019) no. 3, pp. 1251-1337.

Résumé

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.

Détail

DOI : 10.2140/gt.2019.23.1251

Classification : 22E40, 53C07, 14H60, 20H10
Keywords: character varieties, mapping class group, Higgs bundles, maximal representations

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