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Compactifications of character varieties and skein relations on conformal blocks

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Abstract

Let \(M_C(G)\) be the moduli space of semistable principal G-bundles over a smooth curve C. We show that a flat degeneration of this space \(M_{C_{\Gamma }}(G)\) associated to a singular stable curve \(C_{\Gamma }\) contains the free group character variety \({\mathcal {X}}(F_g, G)\) as a dense open subset, where \(g = genus(C).\) In the case \(G = SL_2({\mathbb {C}})\) we describe the resulting compactification explicitly, and in turn we conclude that the coordinate ring of \(M_{C_{\Gamma }}(SL_2({\mathbb {C}}))\) is presented by homogeneous skein relations. Along the way, we prove the parabolic version of these results over stable, marked curves \((C_{\Gamma }, \vec {p}_{\Gamma })\).

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Acknowledgments

We thank Sean Lawton for many useful conversations about free group character varieties, Neil Epstein for his helpful remarks on Rees algebras, Geir Agnarsson for sharing his knowledge of graph theory, and Steven Sam for a helpful discussion on the material in Sect. 9.4. We thank Kaie Kubjas and Nick Early for useful remarks on an earlier version of this manuscript, and the reviewers for suggesting we look at the work of Florentino [13] and Faltings [9].

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  1. Department of Mathematical Sciences, George Mason University, Fairfax, VA, 22030, USA

    Christopher Manon

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Manon, C. Compactifications of character varieties and skein relations on conformal blocks. Geom Dedicata 179, 335–376 (2015). https://doi.org/10.1007/s10711-015-0084-6

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