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Lower bounds of Tian’s invariant under toric invariances. (English) Zbl 1209.32012

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 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 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 [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 J. Song [Am. J. Math. 127, No. 6, 1247–1259 (2005; Zbl 1088.32012)].

MSC:

32Q20 Kähler-Einstein manifolds
53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
14J45 Fano varieties
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