Toric hypersymplectic quotients

TY - JOUR

T1 - Toric hypersymplectic quotients

AU - Dancer, Andrew

AU - Swann, Andrew

PY - 2007/3/1

Y1 - 2007/3/1

N2 - We study the hypersymplectic spaces obtained as quotients of flat hypersymplectic space ℝ4d by the action of a compact Abelian group. These 4n-dimensional quotients carry a multi-Hamilitonian action of an ntorus. The image of the hypersymplectic moment map for this torus action may be described by a configuration of solid cones in ℝ3n. We give precise conditions for smoothness and non-degeneracy of such quotients and show how some properties of the quotient geometry and topology are constrained by the combinatorics of the cone configurations. Examples are studied, including non-trivial structures on ℝ4n and metrics on complements of hypersurfaces in compact manifolds.

AB - We study the hypersymplectic spaces obtained as quotients of flat hypersymplectic space ℝ4d by the action of a compact Abelian group. These 4n-dimensional quotients carry a multi-Hamilitonian action of an ntorus. The image of the hypersymplectic moment map for this torus action may be described by a configuration of solid cones in ℝ3n. We give precise conditions for smoothness and non-degeneracy of such quotients and show how some properties of the quotient geometry and topology are constrained by the combinatorics of the cone configurations. Examples are studied, including non-trivial structures on ℝ4n and metrics on complements of hypersurfaces in compact manifolds.

KW - Hypersymplectic structure

KW - Moment map

KW - Neutral hyperkähler manifold

KW - Toric variety

UR - http://www.scopus.com/inward/record.url?scp=64849093134&partnerID=8YFLogxK

U2 - 10.1090/S0002-9947-06-03925-0

DO - 10.1090/S0002-9947-06-03925-0

M3 - Journal article

AN - SCOPUS:64849093134

SN - 0002-9947

VL - 359

SP - 1265

EP - 1284

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

IS - 3

ER -