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2022 | OriginalPaper | Chapter

Symplectic Resolutions for Multiplicative Quiver Varieties and Character Varieties for Punctured Surfaces

Authors : Travis Schedler, Andrea Tirelli

Published in: Representation Theory and Algebraic Geometry

Publisher: Springer International Publishing

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Abstract

We study the algebraic symplectic geometry of multiplicative quiver varieties, which are moduli spaces of representations of certain quiver algebras, introduced by Crawley-Boevey and Shaw in 2006, called multiplicative preprojective algebras. They are multiplicative analogues of Nakajima quiver varieties. They include character varieties of (open) Riemann surfaces fixing conjugacy class closures of the monodromies around punctures, when the quiver is “crab-shaped.” We prove that under suitable hypotheses on the dimension vector of the representations, or the conjugacy classes of monodromies in the character variety case, the normalizations of such moduli spaces are symplectic singularities and the existence of a symplectic resolution depends on a combinatorial condition on the quiver and the dimension vector. These results are analogous to those obtained by Bellamy and the first author in the ordinary quiver variety case and for character varieties of closed Riemann surfaces. At the end of the paper, we outline some conjectural generalizations to moduli spaces of objects in 2-Calabi–Yau categories.

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Footnotes
1
Since submission of this article, in [KS], the first author and D. Kaplan resolve this in the affirmative for all multiplicative quiver varieties. In particular, they are all normal.
 
2
As mentioned in Sect. 1.1, it is now known that these and the other varieties in this article are normal.
 
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Metadata
Title
Symplectic Resolutions for Multiplicative Quiver Varieties and Character Varieties for Punctured Surfaces
Authors
Travis Schedler
Andrea Tirelli
Copyright Year
2022
DOI
https://doi.org/10.1007/978-3-030-82007-7_9

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