On the Futaki invariant of Fano threefolds

We study the zero locus of the Futaki invariant on K-polystable Fano threefolds, seen as a map from the Kähler cone to the dual of the Lie algebra of the reduced automorphism group. We show that, apart from families 3.9, 3.13, 3.19, 3.20, 4.2, 4.4, 4.7 and 5.3 of the Iskovskikh–Mori–Mukai classification of Fano threefolds, the Futaki invariant of such manifolds vanishes identically on their Kähler cone. In all cases, when the Picard rank is greater or equal to two, we exhibit explicit 2-dimensional differentiable families of Kähler classes containing the anti-canonical class and on which the Futaki invariant is identically zero. As a corollary, we deduce the existence of non Kähler–Einstein cscK metrics on all such Fano threefolds.