Del Pezzo surfaces with many symmetries
The study of Kähler-Einstein metrics on Fano manifolds is a major topic in borderline between differential geometry and algebraic geometry. \textit{Y. A. Rubinstein} [Adv. Math. 218, No. 5, 1526--1565 (2008; Zbl 1143.53065)] studied Ricci iterations and their existence. He proved that the unique Kähler-Einstein form on a compact Kähler manifold \(X\) in certain cases is the limit of Ricci iterations. In a particular situation, a necessary condition for this to hold is that \(\alpha_G(X)>1\), for some compact subgroup \(G\subset \text{Aut}(X)\). The \(G\)-equivariant \(\alpha\)-invariant \(\alpha_G(X)\) is defined to be the infimum of \(\text{lct}_n(X,G)\) over all \(n\in\mathbb{N}\), where \[ \text{lct}_n(X,G)= \text{Sup}\{\lambda\in\mathbb{Q} \text{, where } (X,\frac{\lambda}{n}D)\text{ is log canonical for all \(G\)-equivariant divisors }D\in\mid -nK_X\mid\} \] The autors classify del Pezzo surfaces (Fano manifolds in dimension 2) for which there exists a finite subgroup \(G\subset\text{Aut}(X)\) such that \(\alpha_G(X)>1\).
scientific article; zbMATH DE number 6203573
scientific article