Counterexamples to hyperkahler Kirwan surjectivity

2019/04/26 by McGerty, Kevin, Nevins, Thomas
#Algebraic Geometry (math.AG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1904.12003

Abstract

Suppose that M is a complete hyperkahler manifold with a compact Lie group K acting via hyperkahler isometries and with hyperkahler moment map (μ_ℂ, μ_ℝ): M→ \mathfrakk^*\otimesIm(ℍ). It is a long-standing problem to determine when the hyperkahler Kirwan map H^*_K(M,ℚ) \longrightarrow H^*(M//K, ℚ) is surjective. We show that for each n≥ 2, the natural U(n)-action on M = T^*(SL_n×ℂⁿ) admits a hyperkahler quotient for which the hyperkahler Kirwan map fails to be surjective. As a tool, we establish a ``Kahler = GIT quotient'' assertion for products of cotangent bundles of reductive groups, equipped with the Kronheimer metric, and representations.

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• Surjectivity of the hyperkähler Kirwan map
• On the cohomology of hyperkahler quotients
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• The Kirwan map, equivariant Kirwan maps, and their kernels
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