Abstract:We investigate the relationship between Matsushima's obstruction and the Futaki invariant for the existence of Einstein-Kahler metrics on tone Fano fourfolds.In particular, we determine all toric Fano...We investigate the relationship between Matsushima's obstruction and the Futaki invariant for the existence of Einstein-Kahler metrics on tone Fano fourfolds.In particular, we determine all toric Fano fourfolds with vanishing Futaki invariant.Moreover, we construct a non-trivial example of an Einstein-Kahler toric Fano fourfold. Introduction.Let Y be a Fano r-fold, which is by definition, an r-dimensional compact connected non-singular projective algebraic variety, defined over C, with ample anti-canonical line bundle.Then one can naturally ask whether Y admits an Einstein-Kahler metric.As to such existence of Einstein-Kahler metrics, two obstructions are known (see Matsushima [9] and Futaki [4]).We here consider the following for toric Fano r-folds (see Defmiton 1.1).PROBLEMS.(I r ) Classify all toric Fano r-folds with vanishing Futaki invariant.(II r ) For a toric Fano r-fold Y with vanishing Futaki invariant, is its automorphism group Aut(F) a reductive algebraic group?(IΠ Γ ) Does a toric Fano r-fold with vanishing Futaki invariant always admit an Einstein-Kahler metric?Note that if (IΠ r ) is true, then (Π r ) is also true (see Matsushima [9]).For r^3, (I r ) and (IΠ r ) were settled (see Mabuchi [7], Siu [14], Tian and Yau [15]).By Batyrev's recent classification of toric Fano fourfolds [2], it is now possible to study the above problems for r = 4.In this paper, we give a complete classification for (I 4 ), and answer the question (II 4 ) (see Theorem 3.5).Moreover, we can solve (IΠ 4 ) except in one case (see Theorem 4.1).Thanks are due to Professor Toshiki Mabuchi for introducing the author to this topic and giving helpful suggestions.The author wishes to thank Professor Victor V. Batyrev who made available his new results on the classification of toric Fano fourfolds.Read More