Geometry & Topology

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

Daniele Alessandrini and Brian Collier

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Abstract

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.

Article information

Source
Geom. Topol., Volume 23, Number 3 (2019), 1251-1337.

Dates
Received: 27 August 2017
Accepted: 21 July 2018
First available in Project Euclid: 5 June 2019

Permanent link to this document
https://projecteuclid.org/euclid.gt/1559700274

Digital Object Identifier
doi:10.2140/gt.2019.23.1251

Mathematical Reviews number (MathSciNet)
MR3956893

Zentralblatt MATH identifier
07079059

Subjects
Primary: 22E40: Discrete subgroups of Lie groups [See also 20Hxx, 32Nxx] 53C07: Special connections and metrics on vector bundles (Hermite-Einstein- Yang-Mills) [See also 32Q20]
Secondary: 14H60: Vector bundles on curves and their moduli [See also 14D20, 14F05] 20H10: Fuchsian groups and their generalizations [See also 11F06, 22E40, 30F35, 32Nxx]

Keywords
character varieties mapping class group Higgs bundles maximal representations

Citation

Alessandrini, Daniele; Collier, Brian. The geometry of maximal components of the $\mathsf{PSp}(4,\mathbb R)$ character variety. Geom. Topol. 23 (2019), no. 3, 1251--1337. doi:10.2140/gt.2019.23.1251. https://projecteuclid.org/euclid.gt/1559700274


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