The Nekrasov Conjecture for Toric Surfaces

Homological Mirror Symmetry for Toric del Pezzo Surfaces

We prove the homological mirror conjecture for toric del Pezzo surfaces. In this case, the mirror object is a regular function on an algebraic torus (C×)2. We show that the derived Fukaya category of this mirror coincides with the derived category of coherent sheaves on the original manifold.

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 ...

Special Linear Systems on Toric Varieties

We consider linear systems on toric varieties of any dimension, with invariant base points, giving a characterization of special linear systems. We then make a new conjecture for linear systems on rational surfaces. Introduction Let us take the projective plane P and let us consider the linear system of curves of degree d having some points of fixed multiplicity. The expected dimension of such ...

Fuzzy Toric Geometries

We describe a construction of fuzzy spaces which approximate projective toric varieties. The construction uses the canonical embedding of such varieties into a complex projective space: The algebra of fuzzy functions on a toric variety is obtained by a restriction of the fuzzy algebra of functions on the complex projective space appearing in the embedding. We give several explicit examples for ...

On a Duality in Calogero-Moser-Sutherland Systems