Toric Hyperkahler Varieties

TL;DR

This paper develops a comprehensive theory of toric hyperkahler varieties, connecting toric geometry, matroid theory, and convex polyhedra, and extends classical combinatorial applications to the hyperkahler setting.

Contribution

It introduces a new framework for toric hyperkahler varieties as intersections in Lawrence toric varieties, linking them to matroid theory and extending toric geometry applications.

Findings

Toric hyperkahler varieties share the same cohomology ring as Lawrence toric varieties.

The framework generalizes to toric quiver varieties when the matroid is graphic.

Classical combinatorial theorems like Hard Lefschetz are extended to hyperkahler varieties.

Abstract

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.