Toric Geometry of Convex Quadrilaterals
نویسنده
چکیده
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 ones, and show that for a toric orbi-surface with 4 fixed points of the torus action, the vanishing of the Futaki invariant is a necessary and sufficient condition for the existence of Kähler metric with constant scalar curvature. Our results also provide explicit examples of relative K–unstable toric orbi-surfaces that do not admit extremal metrics.
منابع مشابه
Log geometry and multiplier ideals
I work in combinatorics, algebraic geometry, convex geometry and commutative algebra while staying informed on certain topics in category theory and ring theory. In particular, I focus on toric varieties and singularity theory. The study of toric varieties lies at the intersection of combinatorics, algebraic geometry, convex geometry and integer programming. There is a correspondence between ce...
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