The cohomology of hyperkahler quotients.

Thomas Nevins (UIUC)

Monday 30th April, 2018 16:00-17:00 Maths 311B


Starting from a hyperkahler manifold with an action of a compact Lie group, one can build a hyperkahler quotient as the quotient space of a certain submanifold determined by the hyperkahler structure and the group action.  Many interesting manifolds can be built this way, even starting from a group acting on a quaternionic vector space: for example, moduli spaces of instantons on R^4, Nakajima quiver varieties, moduli spaces of Higgs bundles, and more.  Joint work with McGerty provides generators of the cohomology of Nakajima quiver varieties (and of the pure part of the cohomology for multiplicative quiver varieties).  I will explain where such generators come from, and why general principles in noncommutative algebraic geometry yield these and other results.  Finally, if there is time I will explain some situations in which the methods above break down, and sketch some open problems in the area.  No prior knowledge of either hyperkahler geometry or noncommutative algebraic geometry will be assumed. 

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