East Asian Symplectic Conference 2009
Institute of Mathematics, Academia Sinica, Taipei, Taiwan
On the cohomology of hyperk¨ ahler quotients
Young-Hoon Kiem May 6 - 10, 2009
Department of Mathematical Sciences, Seoul National University, Seoul 151-747, Korea
E-mail:kiem@math.snu.ac.kr Abstract
Hyperk¨ahler manifolds are manifolds M equipped with a Riemannian metric g and three independent complex structures i, j, k compatible with the metric which satisfy ij = k = −ji. They correspondingly have three symplectic forms ω1, ω2, ω3, or one real symplectic form ω1 and one complex symplectic form ωC = ω2 + iω3. Suppose a compact connected Lie group K acts on M preserving the metric and the symplectic forms. We say the action is Hamiltonian if there are moment maps µi for each ωi. It has been an outstanding problem how much of the package of properties of Hamiltonian group actions on symplectic manifolds extends to hyperk¨ahler quotients
M///K := µ−1(0)/K
where µ = (µ1, µ2, µ3). I will show that the partial desingularization construction of Kirwan’s extends to hyperk¨ahler quotients and give a criterion for the surjectivity of an analogue of the Kirwan map to the cohomology of hyperk¨ahler quotients. This criterion is applied to some linear actions on hyperk¨ahler vector spaces. This is a joint work with Lisa Jeffrey and Frances Kirwan.
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