Tian Invariant on the Blown-up Grassmannian Futaki Obstructions and K-Stability

In this paper, we compute the Tian invariant on a generalization of Calabi manifolds introduced by the author in collaboration with A. Ben Abdesselem and M. Filali in [1]. We adapt the method of the original article to the setting of complex Grassmannians blown up along linear subvarieties. Our original contribution is fourfold: we explicitly determine the subgroup H ⊂ Aut(G(k, n)) = P GL(n, C) that preserves the linear subvariety L, and we show that the functions ψj are exactly the functions invariant under this group; we establish a new obstruction to the existence of K¨ahler-Einstein metrics on X in terms of the Tian invariant and the Futaki obstruction; we show that for certain values of k and n, the Tian invariant attains the upper bound α(X) = dim X dim X+1 , which ensures the existence of a K¨ahler-Einstein metric; and we establish an equivalence between the K-stability of X and a simple numerical condition on k and n. These results are new and generalize previous work on Calabi manifolds.