TY - JOUR
T1 - The geometry of maximal components of the PSp(4;ℝ) character variety
AU - Alessandrini, Daniele
AU - Collier, Brian
N1 - Publisher Copyright:
© 2019, Mathematical Sciences Publishers. All rights reserved.
PY - 2019
Y1 - 2019
N2 - 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.
AB - 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.
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U2 - 10.2140/gt.2019.23.1251
DO - 10.2140/gt.2019.23.1251
M3 - Article
AN - SCOPUS:85068846759
SN - 1465-3060
VL - 23
SP - 1251
EP - 1337
JO - Geometry and Topology
JF - Geometry and Topology
IS - 3
ER -