Hausel, TamasIST Austria; Sturmfels, Bernd
Extending work of Bielawski-Dancer 3 and Konno 14, 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 11, are extended to the hyperkähler setting. When the matroid is graphic, our construction gives the toric quiver varieties, in the sense of Nakajima 17.
Both authors were supported by the Miller Institute for Basic Research in Science, in the form of a Miller Research Fellowship (1999-2002) for the first author and a Miller Professorship (2000-2001) for the second author. The second author was also supported by the National Science Foundation (DMS-9970254).
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Hausel T, Sturmfels B. Toric hyperkähler varieties. Documenta Mathematica. 2002;7(1):495-534.
Hausel, T., & Sturmfels, B. (2002). Toric hyperkähler varieties. Documenta Mathematica, 7(1), 495–534.
Hausel, Tamas, and Bernd Sturmfels. “Toric Hyperkähler Varieties.” Documenta Mathematica 7, no. 1 (2002): 495–534.
T. Hausel and B. Sturmfels, “Toric hyperkähler varieties,” Documenta Mathematica, vol. 7, no. 1, pp. 495–534, 2002.
Hausel T, Sturmfels B. 2002. Toric hyperkähler varieties. Documenta Mathematica. 7(1), 495–534.
Hausel, Tamas, and Bernd Sturmfels. “Toric Hyperkähler Varieties.” Documenta Mathematica, vol. 7, no. 1, Deutsche Mathematiker Vereinigung, 2002, pp. 495–534.