Stable cohomology of universal character varieties

2025/12/15 by Ishan Banerjee, Faye Jackson, Banerjee, Ishan +7
Mathematics · #Algebraic Geometry and Number Theory #Advanced Algebra and Geometry #Geometry and complex manifolds

paper · pdf · doi:10.48550/arxiv.2512.13879

Abstract

We study the universal PGL_ncharacter variety over M_g whose fiber over a point [C] is the space of PGL_n-local systems on the curve C. We use nonabelian Hodge theory and properties of Saito's mixed Hodge modules to show that the Leray-Serre spectral sequence for the projection to M_g degenerates at E₂. As an application, we prove that the rational cohomology of these varieties stabilizes as g goes to infinity and compute the stable limit. We also deduce similar results for the universal G-character variety over M_g,1 whose fiber over a punctured curve is the variety of G-local systems with fixed central monodromy around the puncture, for G = GL_n or SL_n.

Citations

• Hitchin fibrations are Ngô fibrations
• The Cohen--Lenstra moments over function fields via the stable homology of non-splitting Hurwitz spaces
• Topological K-theory of quasi-BPS categories for Higgs bundles
• Mixed Hodge modules on stacks
• Combinatorial decomposition theorem for Hitchin systems via zonotopes
• Hodge-to-singular correspondence for reduced curves
• The weights of the tautological classes of character varieties
• Global topology of the Hitchin system
• Topology of Hitchin systems and Hodge theory of character varieties: the case A₁
• The Decomposition Theorem and the topology of algebraic maps
• Mixed Hodge polynomials of character varieties
• The stable moduli space of Riemann surfaces: Mumford's conjecture
• Generators for the cohomology ring of the moduli space of rank 2 Higgs bundles
• The Hodge filtration on nonabelian cohomology
• Stable cohomology of the mapping class group with symplectic coefficients and of the universal Abel-Jacobi map

Related

• Character varieties with Zariski closures of GL_n-conjugacy classes at punctures
• PGL_n(ℂ)-character stacks and Langlands duality over finite fields
• Local curves, wild character varieties, and degenerations
• Cell decompositions of character varieties
• Non vanishing loci of Hodge numbers of local systems