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

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.

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Keywords

character varieties, mapping class group, Higgs bundles, maximal representations

Mathematical Subject Classification 2010

Primary: 22E40, 53C07

Secondary: 14H60, 20H10

References

Publication

Received: 27 August 2017

Accepted: 21 July 2018

Published: 28 May 2019

Proposed: Benson Farb

Seconded: Jean-Pierre Otal, Anna Wienhard

Authors

Daniele Alessandrini
Mathematisches Institut Universitaet Heidelberg Heidelberg Germany

Brian Collier
Department of Mathematics University of Maryland College Park, MD United States

Volume 23, issue 3 (2019)

Daniele Alessandrini and Brian Collier

Abstract