We establish a correspondence between simplicial fans, not necessarily rational, and certain foliated compact complex manifolds called LVMBmanifolds. In the rational case, Meersseman and Verjovsky have shown that the leaf space is the usual toric variety. We compute the basic Betti numbers of the foliation for shellable fans. When the fan is in particular polytopal, we prove that the basic cohomology of the foliation is generated in degree two. We give evidence that the rich interplay between convex and algebraic geometries embodied by toric varieties carries over to our nonrational construction. In fact, our approach unifies rational and nonrational cases.
The purpose of this article is to view the Penrose kite from the perspective
of symplectic geometry.Comment: 24 pages, 7 figures, minor changes in last version, to appear in
Comm. Math. Phys
In this paper we study quaternion-Ka$ hler manifolds endowed with an isometric S "-action. We consider the corresponding moment map µ and prove that the only compact quaternion-Ka$ hler manifold with positive scalar curvature which admits an isometric circle action free on µ −" (0) is the quaternionic projective space P n .
In this article we extend cutting and blowing up to the nonrational symplectic toric setting. This entails the possibility of cutting and blowing up for symplectic toric manifolds and orbifolds in nonrational directions.
We construct, for each convex polytope, possibly nonrational and nonsimple, a
family of compact spaces that are stratified by quasifolds, i.e. each of these
spaces is a collection of quasifolds glued together in an suitable way. A
quasifold is a space locally modelled on $\R^k$ modulo the action of a
discrete, possibly infinite, group. The way strata are glued to each other also
involves the action of an (infinite) discrete group. Each stratified space is
endowed with a symplectic structure and a moment mapping having the property
that its image gives the original polytope back. These spaces may be viewed as
a natural generalization of symplectic toric varieties to the nonrational
setting.Comment: LaTeX, 29 pages. Revised version: TITLE changed, reorganization of
notations and exposition, added remarks and reference
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