Toric hyperkähler varieties

  • Tamás Hausel

  • Bernd Sturmfels

    Department of Mathematics Department of Mathematics University of Texas at University of California at Austin TX 78712, USA Berkeley CA 94720, USA
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Extending work of Bielawski-Dancer citeBD and Konno citeKo, we develop a theory of toric hyperkähler varieties, which involves toric geometry, matroid theory and convex polyhedra. The framework is a detailed study of semi-projective toric varieties, meaning GIT quotients of affine spaces by torus actions, and specifically, of Lawrence toric varieties, meaning GIT quotients of even-dimensional affine spaces by symplectic torus actions. A toric hyperkähler variety is a complete intersection in a Lawrence toric variety. Both varieties are non-compact, and they share the same cohomology ring, namely, the Stanley-Reisner ring of a matroid modulo a linear system of parameters. Familiar applications of toric geometry to combinatorics, including the Hard Lefschetz Theorem and the volume polynomials of Khovanskii-Pukhlikov citeKP, are extended to the hyperkähler setting. When the matroid is graphic, our construction gives the toric quiver varieties, in the sense of Nakajima citeNa.

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Tamás Hausel, Bernd Sturmfels, Toric hyperkähler varieties. Doc. Math. 7 (2002), pp. 495–534

DOI 10.4171/DM/130