Big monodromy for higher Prym representations

Landesman, Aaron; Litt, Daniel; Sawin, Will

doi:10.2140/gt.2025.29.2733

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Big monodromy for higher Prym representations

Aaron Landesman, Daniel Litt and Will Sawin

Geometry & Topology 29 (2025) 2733–2782

DOI: 10.2140/gt.2025.29.2733

Abstract

Let $Σ_{g^{'}} \to Σ_{g}$ be a cover of an orientable surface of genus $g$ by an orientable surface of genus $g^{'}$ , branched at $n$ points, with Galois group $H$ . Such a cover induces a virtual action of the mapping class group ${Mod}_{g, n + 1}$ of a genus $g$ surface with $n + 1$ marked points on $H^{1} (Σ_{g^{'}}, ℂ)$ . When $g$ is large in terms of the group $H$ , we calculate precisely the connected monodromy group of this action. The methods are Hodge-theoretic and rely on a “generic Torelli theorem with coefficients”.

Keywords

monodromy, Prym representations, curves, mapping class groups, Hodge theory

Mathematical Subject Classification

Primary: 14H30

Secondary: 14D07, 14H10, 14H40, 14H60, 57K20

References

Publication

Received: 29 April 2024

Accepted: 19 October 2024

Published: 14 August 2025

Proposed: Benson Farb

Seconded: Dan Abramovich, David Fisher

Authors

Aaron Landesman
Department of Mathematics FAS Harvard University Cambridge, MA United States

Daniel Litt
Department of Mathematics University of Toronto Toronto, ON Canada

Will Sawin
Department of Mathematics Princeton University Princeton, NJ United States

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Volume 29, issue 5 (2025)