Alpha invariants and coercivity of the Mabuchi functional on Fano manifolds (Q332673)

Let \((X,\omega)\) be a Kähler manifold with slope NEWLINENEWLINE\[NEWLINE\mu(X,[\omega]):=\frac{\int_Xc_1(X)\cdot \omega^{n-1}}{\int_X \omega^n}.NEWLINE\]NEWLINENEWLINE\textit{G. Tian} [Canonical metrics in Kähler geometry. Notes taken by Meike Akveld. Basel: Birkhäuser (2000; Zbl 0978.53002)] has conjectured that \((X,\omega)\) admits a constant scalar curvature Kähler metric if and only if the Mabuchi functional is coercive. This is proven by Tian when \(\omega \in c_1(X)\). Define NEWLINENEWLINE\[NEWLINE\mathcal{H}(\omega):=\big\{\phi\in C^\infty(X,\mathbb{R})\mid \omega +i\partial \overline \partial \phi>0\big\},NEWLINE\]NEWLINE and define the \(\alpha\)-invariant NEWLINENEWLINE\[NEWLINE\alpha(X,[\omega]))=\text{sup}\Big\{\beta\;\Big|\; \int_X e^{-\beta(\phi-\text{sup}_X\phi)}\omega^n<c\Big\},NEWLINE\]NEWLINE NEWLINEfor some \(c\) independent of \(\phi \in \mathcal{H}(\omega)\). Similarly, one can define \(\alpha_G(X,[\omega])\) by considering only \(\phi\) which are \(G\)-invariant where \(G\) is a compact subgroup of \(\text{Aut}(X,[\omega])\). Tian has proven that if \(X\) is an \(n\)-dimensional Fano manifold with \(\alpha(X,[\omega])> \frac{n}{n+1}\), then \(X\) admits a Kähler-Einstein metric. This theorem was used by \textit{G. Tian} et al. [Invent. Math. 101, No. 1, 101--172 (1990; Zbl 0716.32019); Commun. Math. Phys. 112, 175--203 (1987; Zbl 0631.53052)] in the classification of Kähler-Einstein metrics on del Pezzo surfaces.NEWLINENEWLINENEWLINEThe main result of the paper under review is the following generalization of Tian's result: suppose NEWLINENEWLINE\[NEWLINE\alpha_G(X,[\omega])>\mu(X,[\omega])\frac{n}{n+1}\quad\text{and} \quad c_1(X)\geq \frac{n}{n+1}\mu(X,[\omega])[\omega],NEWLINE\]NEWLINE then the Mabouchi functional is coercive on the space of \(G\)-invariant \(\phi\in \mathcal{H}(\omega)\). As an application of the main result, it is also proven that the \(\alpha\)-invariant is a continuous function on the Kähler cone. The main theorem can also be used to give explicit Kähler classes with coercive Mabuchi functional.