Combinatorial formulae for Futaki characters and generalized Killing forms of toric Fano orbifolds (Q2752847)

For the existence of Einstein-Kähler metrics on an \(r\)-dimensional compact complex orbifold \(X\) with positive first Chern class two obstructions are known one of which is \textit{Y. Matsushima}'s obstruction [Nagoya Math. J. 46, 161--173 (1972; Zbl 0249.53050)] and the other is the \textit{A. Futaki} character [Invent. Math. 73, 437--443 (1983; Zbl 0506.53030)]. These obstructions are defined on the Lie algebra of holomorphic vector fields.NEWLINENEWLINENEWLINEThe author considers the following two conjectures for \(r\)-dimensional toric Fano orbifolds:NEWLINENEWLINENEWLINEConjecture 1.2: If an \(r\)-dimensional toric Fano orbifold has vanishing Futaki character, then its automorphism group is a reductive algebraic group.NEWLINENEWLINENEWLINEConjecture 1.3: If an \(r\)-dimensional toric Fano orbifold has vanishing Futaki character, then it admits an Einstein-Kähler.NEWLINENEWLINENEWLINEThe author observes that if Conjecture 1.3 is true, then Conjecture 1.2 is also true. Whenever \(X\) is nonsingular and \(r\geq 4\), Conjecture 1.3 for \(X\) was solved affirmatively, except the four-dimensional toric Fano manifold of type \(X-1\), by \textit{Y. Nakagawa} [Tôhoku Math. J., II. Ser. 46, 125-133 (1994; Zbl 0838.32008)] and the author gives a table for non-singular four-dimensional Fano polytopes. Then the author establishes a combinatorial formula for the Futaki characters of a toric Fano orbifold and using this formula and classifications of toric Fano orbifolds given by \textit{V. V. Batyrev} [Izv. Akad. Nauk SSSR, Ser. Mat. 45, 704--717 (1981; Zbl 0478.14032)], \textit{R. Koelman} [Thesis, Univ. Nijmegen, 1991], \textit{H. Sato} [Tôhoku Math. J., II. Ser. 52, No. 3, 383--413 (2000; Zbl 1028.14015)] and \textit{K. Watanabe} and \textit{M. Watanabe} [Tokyo J. Math. 5, 37--48 (1982; Zbl 0581.14028)], the author proves Conjecture 1.2 for two-dimensional toric Fano orbifolds and non-singular toric Fano manifolds of dimension \(r\leq 4\). Moreover, the author also proves Conjecture 1.3 for two-dimensional toric Fano orbifolds. Finally, the author considers for toric Fano orbifolds two further conjectures, inspired by \textit{E. Calabi} [Extremal Kähler metrics. II, in ``Differential geometry and complex analysis'', Springer-Verlag, Berlin etc., 95--114 (1985; Zbl 0574.58006)] via T. Mabuchi.NEWLINENEWLINEFor the entire collection see [Zbl 0953.00035].