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Scalar curvature and stability of toric varieties. (English) Zbl 1074.53059
The author deals with the general problem of finding conditions under which a complex projective variety admits a Kähler metric of constant scalar curvature. He states the following conjecture: A smooth polarised projective variety $$(V,L)$$ admits a Kähler metric of constant scalar curvature in the class $$c_1(L)$$ if and only if it is $$K$$-stable. He begins the investigation of the problem in the special case of toric varieties, working within a general differential-geometric framework developed by Guillemin and Abreu. For any compact Kähler manifold $$(V,\omega_0)$$ there is the Mabuchi functional $$\mathcal{M}$$ defined on the metrics in the class $$[\omega_0]$$, whose critical points are the metrics of constant scalar curvature. The main result of this paper is: Theorem 1.1. If a polarised toric surface is $$K$$-stable then the Mabuchi functional $$\mathcal{M}$$ is bounded below on $$\mathcal{H}^T$$ and any minimizing sequence has a $$K$$-convergent subsequence.

##### MSC:
 53C55 Global differential geometry of Hermitian and Kählerian manifolds 53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions 32Q15 Kähler manifolds 32Q20 Kähler-Einstein manifolds 14M25 Toric varieties, Newton polyhedra, Okounkov bodies
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