Toric geometry of convex quadrilaterals

Details Toric geometry of convex quadrilaterals Toric geometry of convex quadrilaterals

2009-09-24

arxiv.org/abs/0909.4512v3

Journal of Symplectic Geometry 9 (2011), pp. 343-385

Mathematics

Differential Geometry

36 pages; v2: small changes (typos and sign convention); v3: few typos corrected, adjustments in the trapezoid case; to appear

Scientific paper

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 K\"ahler-Einstein and toric Sasaki-Einstein metrics constructed in [6,22,14]. As a byproduct, we obtain a wealth of extremal toric (complex) orbi-surfaces, including K\"ahler-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 K\"ahler metric with constant scalar curvature. Our results also provide explicit examples of relative K-unstable toric orbi-surfaces that do not admit extremal metrics.

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