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.
Cite this article
Tamás Hausel, Bernd Sturmfels, Toric hyperkähler varieties. Doc. Math. 7 (2002), pp. 495–534DOI 10.4171/DM/130