Toric hyperkähler varieties

By Tamás Hausel, Bernd Sturmfels. Documenta Mathematica, 2002

134 citations recorded by the source corpus.

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.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/dfd599e91e1caa9589d6312a4ea94c8a8ca52d5b

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Citations in the corpus

Multigraded Castelnuovo-Mumford Regularity (2003, unknown): “By [24], Theorem 2.”
On the shape of a pure O-sequence (2010, background): “Most of the huge amount of work done on matroids over the last thirty-four years, involving ideas and techniques coming from several different disciplines, has in fact been motivated by that intriguing conjecture, which remains wide open today (see, for instance, [36, 51, 60, 61, 63, 66, 71, 76, 77]”
Polyhedral divisors and algebraic torus actions (2003, background): “The following class of varieties will be of special importance for us; it comprises the affine as well as the projective ones, compare also [12].”
Mixed Hodge polynomials of character varieties (2006, background): “See also [39] for the original argument on toric hyperkähler varieties.”
Gale duality and Koszul duality (2008, background): “Note that the cohomology of the hypertoric variety MH is independent of both parameters and [Kon00, HS02 , Pro08].”
A survey of hypertoric geometry and topology (2007, background): “6.7], but the following combinatorial interpretation was first observed by [ HSt , 1.2].”
Zonotopal algebra (2007, background): “A nice connection between the space D(X) and the cohomology of toric hyperKähler varieties is described in [28] via the so-called Volume Polynomials shown to span D(X) as a D-module (for subsequent developments, see [27], [29]).”
EQUIVARIANT CHOW COHOMOLOGY OF TORIC VARIETIES (2005, methodology): “We briefly recall the construction of general hypertoric varieties (by “general” we mean that we consider only regular values of the moment map), and refer the reader to [BD], [ HS ], and [PW] for details and further references, as well as for the construction of hypertoric varieties associated to n”
Hodge cohomology of gravitational instantons (2002, background): “An algebraic geometric account of the underlying varieties, with some novel applications to combinatorics, is given in [40].”
Toric reduction and a conjecture of Batyrev and Materov (2003, background): “In this section, A is any projective sequence in Γ∗a. Recall that we have chosen an orientation of a, and that this, together with the lattice Γa induces a volume form dµaΓ on a. Pick a chamber c ⊂ Cone(A) and consider the orbifold toric variety VA(c).”