Kähler-Einstein metrics on smooth Fano toroidal symmetric varieties of type AIII (Q6553560)

A Kähler metric on a complex manifold is said to be Kähler-Einstein if \(Ric(\omega) = \lambda \omega\), where \(\lambda\in \mathbb{R}\) and \(\mathrm{Ric}(\omega)\) denotes the Ricci curvature form of the associated Kähler form \(\omega\). According to the Yau-Tian-Donaldson conjecture a Fano manifold admits a Kähler-Einstein metric if and only if it is \(K\)-polystable. This notion of stability is of an algebro-geometric nature and has its origin in Geometric Invariant Theory. It was introduced by Tian and in its most general form, is due to Donaldson. It is formulated in terms of polarized \(\mathbb{C}^*\)-equivariant deformations \(\mathcal{L}\rightarrow \mathcal{X} \rightarrow \mathbb{C}\) of \((X,-K_X)\) called test configurations (where \(-K_X\) is the anticanonical line bundle and \(\mathcal{X}_1 = X\)). To any test configuration \((\mathcal{X},\mathcal{L})\) one associates a numerical invariant \(DF(\mathcal{X},\mathcal{L})\), called the Donaldson-Futaki invariant and defined in terms of \((\mathcal{X}_0,\mathcal{L}_{|\mathcal{X}_0} )\). \(X\) is said to be \(K\)-polystable if \(DF(\mathcal{X},\mathcal{L})\geq 0\) with equality if and only if (\(\mathcal{X},\mathcal{L}\)) is isomorphic to a product test configuration.\N\NHowever, it is a difficult problem to determine \(K\)-polystability of a given Fano manifold. Fortunately, more manageable \(K\)-polystability criterion has been found for Fano varieties equipped with algebraic group actions. For instance, the works [\textit{T. Mabuchi}, Osaka J. Math. 24, 705--737 (1987; Zbl 0661.53032)] together with [\textit{X.-J. Wang} and \textit{X. Zhu}, Adv. Math. 188, No. 1, 87--103 (2004; Zbl 1086.53067)] imply that a toric Fano manifold is \(K\)-polystable if and only if the barycenter of the moment polytope is the origin.\N\NThis is largely generalized to smooth Fano spherical varieties in terms of its moment polytope and spherical data by [\textit{T. Delcroix}, Ann. Sci. Éc. Norm. Supér. (4) 53, No. 3, 615--662 (2020; Zbl 1473.14098)]. More precisely, a smooth Fano spherical variety \(X\) is K-polystable if and only if the barycenter of the moment polytope of \(X\) with respect to the Duistermaat-Heckman measure belongs to certain cone. Here, a normal variety is called spherical if it admits an action of a reductive algebraic group whose Borel subgroup acts with an open orbit on the variety.\N\NIf a Fano spherical manifold is homogeneous then it is \(K\)-polystable by \textit{Y. Matsushima} [Nagoya Math. J. 46, 161--173 (1972; Zbl 0249.53050)]. There are two additional well-known classes of spherical varieties: symmetric varieties and horospherical varieties. Symmetric varieties are normal equivariant open embeddings of symmetric homogeneous spaces. It is known that non-homogeneous Fano symmetric manifold of Picard number one are \(K\)-polystable, while non-homogeneous Fano horospherical manifold are not. The authors are interested in determining \(K\)-polystability of non-homogeneous Fano symmetric manifolds of Picard number two or higher in the case the open orbit is of type \(AIII(2,m)\).\N\NMore precisely, they consider the open embedding of a symmetric homogeneous space \(G/N_G(G^\theta)\) (where \(\theta\) is an involution of \(G\)) which dominates the wonderful compactification of \(G/N_G(G^\theta)\), i.e. the maximal smooth equivariant compactification having a unique closed \(G\)-orbit (see [\textit{C. De Concini} and \textit{C. Procesi}, Lect. Notes Math. 996, 1--44 (1983; Zbl 0581.14041)].\N\NIt is known that if a Fano symmetric variety is obtainable from a wonderful compactification by a sequence of blow-ups along closed orbits, then it is either \(S\) or the blow-up of \(S\) along the unique closed orbit (see [\textit{A. Ruzzi}, Publ. Res. Inst. Math. Sci. 48, No. 2, 235--278 (2012; Zbl 1245.14042)]). Therefore the authors ask themselves two questions: 1) Are the wonderful compactifications \(K\)-polystable? 2) Which blow-up of wonderful compactifications along the (unique) closed orbit are \(K\)-polystable?\N\NLet \(X_m\) be the wonderful compactifications of symmetric homogeneous spaces of type \(AIII(2,m)\) for \(m\geq 4\) and let \(Y_m\) be its blow-up along the closed orbit. It is known that \(X_m\) is Fano for each \(m\geq 4\) while \(Y_m\) is Fano if \(m\geq 5\) and Calabi-Yau if \(m = 4\). In the case of \(m=4\), the symmetric homogeneous space is \(\mathrm{SL}_4(\mathbb{C}/N_{\mathrm{SL}_4(\mathbb{C}}(S(\mathrm{GL}_2(\mathbb{C}\times \mathrm{GL}_2(\mathbb{C})))\). For \(m\geq 5\), \(X_m\) is the compactification of \(\mathrm{SL}_m(\mathbb{C}/S(\mathrm{GL}_2(\mathbb{C}\times \mathrm{GL}_{m-2}(\mathbb{C}))\) and can be described as the blow-up of the product \(\mathrm{Gr}(2,m)\times \mathrm{Gr}(m-2,m)\) of Grassmannians along the closed orbit \(\mathrm{Fl}(2,m-2;m)\).\N\NThe first main result of this article is that \(X_m\) is \(K\)-polystable for each \(m\geq 4\).\N\NThe second one says that \(Y_m\) is \(K\)-polystable for \(m=4,5\) and \(K\)-unstable for \(m\geq 6\).