Lower bounds of Tian's invariant under toric invariances (Q1957949)

This paper studies the geometry of Kähler potentials on a family of high-dimensional Fano manifolds. More precisely, the authors consider some explicit Fano manifolds which are projective bundles over a complex projective space, which generalize the blow up of complex projective space at one point and also generalize a famous construction of \textit{E. Calabi} [Ann. Math. Stud. 102, 259--290 (1982; Zbl 0487.53057)]. For each such manifold, they fix a finite group \(G\) of automorphisms and they give a lower bound for \textit{G. Tian}'s \(\alpha\)-invariant [Invent. Math. 89, 225--246 (1987; Zbl 0599.53046)] relative to \(G\). This is achieved via a careful study of \(G\)-invariant almost plurisubharmonic functions on these manifolds by explicitly constructing a function bounding below all of these, following the methods developed in [\textit{A. Ben Abdesselem}, Bull. Sci. Math. 130, 341--353 (2006; Zbl 1103.53042)]. Note that the \(\alpha\)-invariant of all toric Fano manifolds has been computed by \textit{J. Song} [Am. J. Math. 127, No.~6, 1247--1259 (2005; Zbl 1088.32012)].