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arXiv · 2512.13879

Stable cohomology of universal character varieties

Ishan Banerjee·Faye Jackson·Anne Larsen·Sam Payne·Xiyan Zhong
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Abstract

We study the universal PGL_n character 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 E_2. 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. The paper concludes with a computation of the ring structure on the stable cohomology and with an appendix by Anne Larsen and Mirko Mauri proving related results for the intersection cohomology of singular universal character varieties.

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Ishan Banerjee, Faye Jackson, Anne Larsen, Sam Payne, Xiyan Zhong. 2025-12-15. Stable cohomology of universal character varieties. https://arxiv.org/abs/2512.13879

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