Toric geometry of cuts and splits

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...

Toric Geometry and String Theory

Acknowledgements First and foremost, I would like to thank Philip Candelas for his invaluable help throughout my four years in Oxford. I also owe special thanks to Bogdan Florea, Marcos Mariño and Harald Skarke for stimulating and enjoyable collaborations on the research topics discussed in this thesis. many others for interesting mathematical physics discussions. Many thanks to my parents, who...

Flag Manifolds and Toric Geometry

1.1. Flag Manifolds. One of the best understood examples of algebraic varieties is the flag manifold. One is first interested in flag varieties as they are well-defined examples for many basic concepts. Besides this, however, one can find that they deserve individual attention, for they naturally arise in many circumstances, e.g. in the theory of characteristic classes of vector bundles, mirror...

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 ...