The real solutions to a system of sparse polynomial equations may be realized as a fiber of a projection map from a toric variety. When the toric variety is orientable, the degree of this map is a lower bound for the number of real solutions to the system of equations. We strengthen previous work by characterizing when the toric variety is orientable. This is based on work of Nakayama and Nishi...

Foreword These notes cover a short course on symplectic toric manifolds, delivered in six lectures at the summer school on Symplectic Geometry of Integrable Hamiltonian Systems, mostly for graduate students, held at the Centre de Recerca Matemàtica in Barcelona in July of 2001. The goal of this course is to provide a fast elementary introduction to toric manifolds (i.e., smooth toric varieties)...

Posterior Chamber Phakic Toric Implantable Collamer Lenses have become increasingly used to correct refractive error associated with astigmatism. These devices are claimed to provide high efficacy in terms of refractive correction. This book chapter is an updated review on the safety and effectiveness and potential complications of the toric implantable collamer lens (Toric ICL) published in pe...

A principal toric bundleM is a complex manifold equipped with a free holomorphic action of a compact complex torus T . Such a manifold is fibered over M/T , with fiber T . We discuss the notion of positivity in fiber bundles and define positive toric bundles. Given an irreducible complex subvariety X ⊂ M of a positive principal toric bundle, we show that either X is T -invariant, or it lies in ...

We define projective GIT quotients, and introduce toric varieties from this perspective. We illustrate the definitions by exploring the relationship between toric varieties and polyhedra. Geometric invariant theory (GIT) is a theory of quotients in the category of algebraic varieties. Let X be a projective variety with ample line bundle L, and G an algebraic group acting on X, along with a lift...

Let O be a symplectic toric orbifold with a fixed T-action and with a toric Kähler metric g. In [10] we explored whether, when O is a manifold, the equivariant spectrum of the Laplace operator ∆g on C∞(O) determines the moment polytope of O, and hence by Delzant’s theorem determines O up to symplectomorphism. In the setting of toric orbifolds we significantly improve upon our previous results a...

Clinical outcomes were compared between high-cylinder toric intraocular lens (IOL) implantation and the combined surgery of low-cylinder toric IOL implantation and limbal relaxing incision (LRI) for correcting preexisting high-amplitude corneal astigmatism. Fifty-seven eyes with preexisting corneal astigmatism of 2.5 diopter (D) or greater were divided into the following two groups: (1) eyes th...

Normal toric varieties over a field or a discrete valuation ring are classified by rational polyhedral fans. We generalize this classification to normal toric varieties over an arbitrary valuation ring of rank one. The proof is based on a generalization of Sumihiro’s theorem to this non-noetherian setting. These toric varieties play an important role for tropicalizations. 2010 Mathematics Subje...

Given a root system, the Weyl chambers in the co-weight lattice give rise to a real toric variety, called the real toric variety associated to the Weyl chambers. We compute the integral cohomology groups of real toric varieties associated to the Weyl chambers of type Cn and Dn, completing the computation for all classical types.

We present a new method to achieve an embedded desingularization of a toric variety. Let W be a regular toric variety defined by a fan Σ and X ⊂ W be a toric embedding. We construct a finite sequence of combinatorial blowing-ups such that the final strict transforms X ′ ⊂ W ′ are regular and X ′ has normal crossing with the exceptional divisor.