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Hodge theory of the Goldman bracket

Richard Hain

Geometry & Topology 24 (2020) 1841–1906

We show that, after completing in the I–adic topology, the Goldman bracket on the space spanned by the closed geodesics on a smooth, complex algebraic curve X is a morphism of mixed Hodge structures. We prove similar statements for the natural action of the loops in X on paths from one boundary vector to another.

Dedicated to the memory of Stefan Papadima

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Goldman bracket, Hodge theory
Mathematical Subject Classification 2010
Primary: 17B62, 58A12
Secondary: 57N05, 14C30
Received: 8 January 2019
Revised: 8 October 2019
Accepted: 26 November 2019
Published: 10 November 2020
Proposed: Dan Abramovich
Seconded: Ulrike Tillmann, Frances Kirwan
Richard Hain
Department of Mathematics
Duke University
Durham, NC
United States