Kähler-Einstein metrics on a Kähler manifold are the solutions of a highly non linear second order partial differential equation on the manifold. A major advance to understand the conditions for the existence of such solution is the resolution of the Yau-Tian-Donaldson conjecture in the Fano case by X. Chen et al. [J. Am. Math. Soc. 28, No. 1, 235–278 (2015; Zbl 1311.53059); J. Am. Math. Soc. 28, No. 1, 199–234 (2015; Zbl 1312.53097); J. Am. Math. Soc. 28, No. 1, 183–197 (2015; Zbl 1312.53096)]. This conjecture states that the existence of some canonical metrics on a Kähler manifold should be related to the algebro-geometric condition of \(K\)-stability on the manifold.
The \(K\)-stability condition involves the positivity of numerical invariants associated to polarized one parameter degenerations of the manifold, equipped with an action of \(\mathbb{C}^*\), called test configurations. In the Fano case, it was proved in [C. Li and C. Xu, Ann. Math. (2) 180, No. 1, 197–232 (2014; Zbl 1301.14026)] that it is enough to consider special test configurations, i.e. test configurations with normal central fiber.
[V. Datar and G. Székelyhidi, Geom. Funct. Anal. 26, No. 4, 975–1010 (2016; Zbl 1359.32019)] gives another proof of the previous conjecture which is relevant to this work. Indeed such proof takes into account automorphisms of the manifold, by considering only equivariant test configurations, and extends the result to Kähler-Ricci solitons.
The author obtains a combinatorial criterion for the existence of Kähler-Einstein metrics on a Fano spherical manifold, involving only the moment polytope and the valuation cone of the spherical manifold. Such a criterion was already knew for some classes of spherical manifolds: toric varieties, generalized flag manifolds, homogeneous toric bundles and biequivariant compactifications of reductive groups. The result of this work generalises such criterions. Horospherical varieties and symmetric varieties are example of spherical varieties for which the result of this work is new.
Given a \(G\)-spherical Fano manifold \(X\), the moment polytope can be characterized from a symplectic point of view as the Kirwan moment polytope of \((X,\omega)\) with respect to the action of a maximal compact subgroup \(K\) of \(G\), where \(\omega\) is a \(K\)-invariant Kähler form in \(c_1(X)\).
The valuation cone of \(X\) depends only on the open orbit of \(X\).
The intuition for the main result comes from author’s previous work on group compactifications, which did not involve \(K\)-stability. The proof of a Kähler-Einstein criterion for smooth and Fano group compactifications in [T. Delcroix, Geom. Funct. Anal. 27, No. 1, 78–129 (2017; Zbl 1364.32017)] can be adapted to provide another proof of the criterion for Kähler-Ricci solitons on the same manifolds.
The computation of the \(K\)-stability of a manifold requires two ingredients. The first one is a description of all (special) test configurations and the second one is a way to compute the Donaldson-Futaki invariant for all of these test configurations. The description of special equivariant test configurations is obtained using the theory of colored fan (such theory permits to classify the \(G\)-spherical varieties with \(G\) fixed). For special test configuration, the central fiber itself is a spherical variety and the action of \(\mathbb{C}^*\) on the central fiber may be deduced from the colored fan of the test configuration.
The author’s description of the action of \(\mathbb{C}^*\) on the central fiber of a special equivariant test configuration relies on the work of M. Brion and F. Pauer [Comment. Math. Helv. 62, 265–285 (1987; Zbl 0627.14038)] on elementary embeddings of spherical homogeneous spaces. The Donaldson-Futaki invariant of a test configuration depends only on the central fiber and the induced action of \(\mathbb{C}^*\). The basic idea of the main result is that one can degenerate the central fiber even more in order to acquire more symmetries, then compute the Futaki invariant on the corresponding degeneration.
For spherical varieties, this idea leads to consider only the Futaki invariants of horospherical varieties. Indeed, there always exist a test configuration with horospherical central fiber. The existence of a horospherical degeneration for spherical varieties is a classical result (see [M. Brion, Manuscr. Math. 55, 191–198 (1986; Zbl 0604.14048); V. L. Popov, Math. USSR, Sb. 58, 311–335 (1987; Zbl 0627.14033)]).
The results hold also for modified \(K\)-stability and existence of Kähler-Ricci solitons.