Toric Hyperkahler Varieties

Journal article

Extending work of Bielawski-Dancer and Konno, we develop a theory of toric hyperkahler 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 hyperkahler 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, are extended to the hyperkahler setting. When the matroid is graphic, our construction gives the toric quiver varieties, in the sense of Nakajima.

Authors: Hausel, T; Sturmfels, B

• Journal: Documenta Mathematica
• Volume: 7
• Issue: 1
• Pages: 495-534
• Publication date: 2002-03-11
• ISSN: 1431-0635
• Language: English
• Keywords: math.CO; math.DG; math.AG