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

Daniele Alessandrini and Brian Collier

Geometry & Topology 23 (2019) 1251–1337

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

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, ) and Sp(4, ), 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, ) and Sp(4, ) 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, )–representations.

character varieties, mapping class group, Higgs bundles, maximal representations
Mathematical Subject Classification 2010
Primary: 22E40, 53C07
Secondary: 14H60, 20H10
Received: 27 August 2017
Accepted: 21 July 2018
Published: 28 May 2019
Proposed: Benson Farb
Seconded: Jean-Pierre Otal, Anna Wienhard
Daniele Alessandrini
Mathematisches Institut
Universitaet Heidelberg
Brian Collier
Department of Mathematics
University of Maryland
College Park, MD
United States