On the modified futaki invariant of complete intersections in projective spaces

Let M be a Fano manifold. We call a Kähler metric ω ∈ c₁(M) a Kähler-Ricci soliton if it satisfies the equation Ric(ω) - ω = L_V ω for some holomorphic vector field V on M. It is known that a necessary condition for the existence of Kähler-Ricci solitons is the vanishing of the modified Futaki invariant introduced by Tian and Zhu. In a recent work of Berman and Nyström, it was generalized for (possibly singular) Fano varieties, and the notion of algebrogeometric stability of the pair (M, V ) was introduced. In this paper, we propose a method of computing the modified Futaki invariant for Fano complete intersections in projective spaces.