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Hyperpolygon spaces, Hitchin systems, and resolutions of finite quotient singularities

Algebraic Geometry

Speaker: Steven Rayan, University of Saskatchewan
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Location: 2112 MSB
Start time: Wed, Feb 22 2023, 4:10PM

Hyperpolygon spaces are Nakajima quiver varieties for the star-shaped quiver. One of the interesting features of such spaces is their connection to Hitchin systems: each hyperpolygon space endows a subvariety of an appropriate meromorphic Hitchin system on the projective line with a complete hyperkähler structure distinct from the Hitchin one. One can use natural invariant functions on the quiver variety to elicit sub-integrable systems within the Hitchin system, as shown in joint work first with J. Fisher and then later with L. Schaposnik. At the same time, certain hyperpolygon spaces can be realized as crepant resolutions of finite quotient singularities. In fact, for certain quotients, all of the crepant resolutions are, remarkably, hyperpolygon spaces, as recently shown in joint work with A. Craw, G. Bellamy, T. Schedler, and H. Weiss. I will describe hyperpolygon spaces from these various perspectives in hopes of demonstrating that they are an inspiring testing ground for questions about the singular and birational geometry of the Hitchin integrable system and other related moduli spaces.

Professor Rayan will also give an informal preparatory seminar on this topic on Tuesday, February 21st, at 10am. Place TBD. Please let Motohico know if you are interested.