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Mathematics, Algebraic Geometry, Rings and Algebras, Representation Theory, Symplectic Geometry
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, 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 normalisations of such moduli spaces are symplectic
singularities and that 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 generalisations to moduli
spaces of objects in 2-Calabi--Yau categories.
