Toric geometry of convex quadrilaterals. (English) Zbl 1233.14032

Let us consider a complex polarized toric manifold. M. Abreu has found a simple expression for the scalar curvature of a toric Kähler metric on such a manifold, in terms of the symplectic potential on the associated moment (Delzant) polytope. It allows to write the constant scalar curvature equation (or more generally the extremal Kähler equation) as a second order ODE in the inverse Hessian of the symplectic potential and an affine function. On the other hand, S.K. Donaldson has defined a notion of (relative) K-stability associated to a given polytope and toric degenerations. This algebraic notion of K-stability is conjectured to be equivalent to the existence of a solution to the extremal Kähler equation.
In the paper under review, the author considers a special type of polytope: a labeled convex quadrilateral. The author shows the existence of a compatible Kähler metric admitting a Hamiltonian 2-form on any symplectic toric orbifold whose moment polytope is a quadrilateral. The interest of such forms is that it is possible to separate variables in the extremal Kähler equation and then to solve this equation explicitly in certain cases. The proofs involved are based on the delicate techniques developed by V. Apostolov, D. Calderbank, P. Gauduchon [“Hamiltonian 2-forms in Kähler geometry. I: General theory”, J. Differ. Geom. 73, No. 3, 359–412 (2006; Zbl 1101.53041)] on such Hamiltonan 2-forms. This proves that the conjecture holds for quadrilaterals and certains affine functions. Previously, S. K. Donaldson had proved the conjecture by a different method for any polytope of dimension 2 but only when the affine function is constant (see [“Constant scalar curvature metrics on toric surfaces”, Geom. Funct. Anal. 19. No. 1, 83–136 (2009; Zbl 1177.53067); “Scalar curvature and stability of toric varieties”, J. Differ. Geom. 62, No. 2, 289–349 (2002; Zbl 1074.53059)]).
Several geometric applications and examples are given. For instance, one can deduce a complete classification of certain Kähler-Einstein and toric Sasaki-Einstein metrics studied in other works and that are interesting for physicists. For convex quadrilaterals \(\Delta\) with vertices that lie in a lattice, it implies existence of a Kähler-Einstein toric orbifold with associated polytope \(\Delta\). An example of relative K-unstable toric orbi-surface that do not admit extremal metric is also provided.
The paper is well written.


14M25 Toric varieties, Newton polyhedra, Okounkov bodies
53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
32Q15 Kähler manifolds
32Q20 Kähler-Einstein manifolds
53D20 Momentum maps; symplectic reduction
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