Toric geometry of G2–manifolds

Branes and Toric Geometry

We show that toric geometry can be used rather effectively to translate a brane configuration to geometry. Roughly speaking the skeletons of toric space are identified with the brane configurations. The cases where the local geometry involves hypersurfaces in toric varieties (such as P blown up at more than 3 points) presents a challenge for the brane picture. We also find a simple physical exp...

The Geometry of Toric Varieties

Toric Geometry of Convex Quadrilaterals

We provide an explicit resolution of the Abreu equation on convex labeled quadrilaterals. This confirms a conjecture of Donaldson in this particular case and implies a complete classification of the explicit toric Kähler– Einstein and toric Sasaki–Einstein metrics constructed in [6, 23, 14]. As a byproduct, we obtain a wealth of extremal toric (complex) orbi-surfaces, including Kähler–Einstein ...

Quiver Representations in Toric Geometry

This article is based on my lecture notes from summer schools at the Universities of Utah (June 2007) and Warwick (September 2007). We provide an introduction to explicit methods in the study of moduli spaces of quiver representations and derived categories arising in toric geometry. The first main goal is to present the noncommutative geometric approach to semiprojective toric varieties via qu...

Some Examples in Toric Geometry

We present two examples in toric geometry concerning the relationship between smooth toric varieties and quasitoric manifolds (or more generally unitary torus manifolds), and extend the results of [8] to prove the non-existence of almost complex quasitorics over the duals of some certain cyclic 4-polytopes. We also provide the sufficient conditions on the base polytope and the characteristic ma...