Alpha invariant and K-stability of \(\mathbb Q\)-Fano varieties
G. Tian introduced in the 80's a holomorphic invariant, the so-called \(\alpha\) invariant, to obtain a sufficient condition for existence of a Kähler-Einstein metric on a given Fano smooth complex manifold \(X\). This condition can simply be written as \(\alpha(X)>n/(n+1)\) where \(n\) is the complex dimension of \(X\). On another hand, it is expected (by the famous Yau-Tian-Donaldson conjecture) that the K-stability of the couple \((X,-K_X)\) should be equivalent to the existence of a Kähler-Einstein metric in the class \(c_1(X)>0\). Therefore it is very natural to expect that one can obtain from Tian's condition the K-stability of \((X,-K_X)\) by a direct algebro-geometric method. In a nutshell, this is what is done by the authors. It is important to remark that their method generalizes to the case of varieties over a closed field of characteristic 0.NEWLINENEWLINEIn this note, the authors prove that if the global log canonical threshold of \(X\) satisfies NEWLINE\[NEWLINElct(X)=\inf_{m\in \mathbb{N}^*} \inf_{D\sim (-mK_X)} \mathrm{lct}(X,\frac{1}{m}D) >\frac{n}{n+1},NEWLINE\]NEWLINE where \(X\) is a log-canonical \(\mathbb{Q}\)-Fano variety, then \((X,-K_X)\) is K-stable. Note that above, \(D\) varies in the set of effective divisors linearly equivalent to \(-mK_X\) and for smooth manifolds, \(\mathrm{lct}(X)=\alpha(X)\) by a result of J-P. Demailly. NEWLINENEWLINENEWLINE NEWLINEThe proof is based on the one hand on a formula for the Donaldson-Futaki invariant derived from the work of the first author and on the other hand from the relationship between the log-canonical thresholds and the Seshadri constants. A \(G\)-invariant version of the main result is given, where \(G\) is a subgroup of the automorphism group \(\mathrm{Aut}(X)\).NEWLINENEWLINEThe paper is well written.
scientific article; zbMATH DE number 6023755
scientific article