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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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