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K-stability of Gorenstein Fano group compactifications with rank two. (English) Zbl 1508.14060

The Yau-Tian-Donaldson conjecture for the Fano case states that a Fano manifold admits a Kähler-Einstein metric if and only if it satisfies the algebraic geometric condition, called the \(K\)-polystability. This conjecture for the Fano case were completely solved by Chen-Donaldson-Sun and Tian. Recently, C. Li [J. Reine Angew. Math. 733, 55–85 (2017; Zbl 1388.53076)] generalized this conjecture to \(\mathbb{Q}\)-Fano varieties with singular Kähler-Einstein metrics using the notion of uniform \(K\)-stability. In general, however, it is difficult to verify the \(K\)-stability condition to show the existence of a Kähler-Einstein metric since one should consider infinite number of possible degenerations (test configurations). When the manifold has large symmetry, it is possible to reduce the problem to checking only finite number of degenerations. In fact, V. Datar and G. Székelyhidi [Geom. Funct. Anal. 26, No. 4, 975–1010 (2016; Zbl 1359.32019)] proved that if a Fano manifold X is equivariant K-stable, i.e., K-stable with respect to special degenerations that are \(G\)-equivariant for some reductive subgroup \(G\) of Aut\(^0 (X)\), then it admits a Kähler-Einstein metric.
By the theorem of X.-J. Wang and X. Zhu [Adv. Math. 188, No. 1, 87–103 (2004; Zbl 1086.53067)] and the theorem of T. Mabuchi [Osaka J. Math. 24, 705–737 (1987; Zbl 0661.53032)] a toric Fano manifold admits a Kähler-Einstein metric if and only if the barycenter of the moment polytope locates at the origin.
Delcroix proved that: i) a smooth Fano group compactification admits a Kähler metric if and only if the barycenter of the corresponding moment polytope translated by \(-2\rho\) locates in the relative interior of the cone generated by positive roots, where \(-2\rho\) denotes the sum of all positive roots of \(G\) (see [T. Delcroix, Geom. Funct. Anal. 27, No. 1, 78–129 (2017; Zbl 1364.32017)]); ii) \(\mathbb{Q}\)-Fano spherical varieties, this combinatorial condition is equivalent to the equivariant \(K\)-stability (see [T. Delcroix, Ann. Sci. Éc. Norm. Supér. (4) 53, No. 3, 615–662 (2020; Zbl 1473.14098)]). The first results was generalized to \(\mathbb{Q}\)-Fano group compactifications in [Y. Li et al., Math. Eng. 5, No. 2, 1–43 (2023)].
Combining these two results the authors gave a combinatorial criterion for the existence of a singular Kähler-Einstein metric on a bi-equivariant compactification of reductive complex Lie group \(G\).
The authors restrict themselves to the case of Gorenstein \(\mathbb{Q}\)-Fano varieties, i.e. normal projective varieties \(X\) with at most Gorenstein singularities such that the anti-canonical divisor \(-K_X\) is an ample \(\mathbb{Q}\)-Cartier divisor. Gorenstein Fano toric surfaces correspond to 16 equivalence classes of reflexive lattice polygons in the plane up to lattice equivalence. The authors generalize this result classifying all Gorenstein Fano bi-equivariant compactifications of semisimple complex Lie groups with rank two, and determining which of them are equivariant \(K\)-stable and admit (singular) Kähler-Einstein metrics.
One advantage of their combinatorial approach is that they can explicitly compute the value of the greatest Ricci lower bound \(R(X)\) (also called Tian’s \(\beta\)-invariant), which is a measurement how far the Fano manifold \(X\) is from being \(K\)-stable. This gives them three new examples on which each solution of the Kähler-Ricci flow is of type II.

MSC:

14M27 Compactifications; symmetric and spherical varieties
32Q20 Kähler-Einstein manifolds
32M12 Almost homogeneous manifolds and spaces
53C55 Global differential geometry of Hermitian and Kählerian manifolds
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