Abstract
For algebraic varieties defined by hyperkähler or, more generally, algebraic symplectic reduction, it is a long-standing question whether the “hyperkähler Kirwan map” on cohomology is surjective. We resolve this question in the affirmative for Nakajima quiver varieties. We also establish similar results for other cohomology theories and for the derived category. Our proofs use only classical topological and geometric arguments.
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Notes
We have in mind Grojnowski’s equivariant elliptic cohomology [9], since it seems to be the only theory currently documented; though the same arguments apply to any theory with standard formal properties.
This is true when one of X, Y is a Nakajima quiver variety: Nakajima proves that the cohomology of a quiver variety is free abelian.
This is, however, abstractly clear: the Chern classes are pulled back along the composite \(X\times \widetilde{Y}\rightarrow BGL({\text {rk}}({\mathcal E}_j^\ell ))\times BGL({\text {rk}}(\widetilde{{\mathcal F}}_j^\ell ))\xrightarrow {\otimes } BGL\big ({\text {rk}}({\mathcal E}_j^\ell ) \cdot {\text {rk}}(\widetilde{{\mathcal F}}_j^\ell )\big )\), hence are polynomials in the cohomology classes generating \(H^*\big (BGL({\text {rk}}({\mathcal E}_j^\ell ))\times BGL({\text {rk}}(\widetilde{{\mathcal F}}_j^\ell ))\big )\).
We thank J. Weyman for help with references.
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Acknowledgements
We thank G. Bellamy for comments on a draft. Kevin McGerty was supported by EPSRC Programme Grant EI/I033343/1. Thomas Nevins was supported by NSF Grants DMS-1159468 and DMS-1502125.
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McGerty, K., Nevins, T. Kirwan surjectivity for quiver varieties. Invent. math. 212, 161–187 (2018). https://doi.org/10.1007/s00222-017-0765-x
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DOI: https://doi.org/10.1007/s00222-017-0765-x