Kähler Ricci flow with vanished Futaki invariant

We study the convergence of the Kähler-Ricci flow on a compact Kähler manifold $(M,J)$ with positive first Chern class $c_1(M;J)$ and vanished Futaki invariant on $πc_1(M;J)$. As the application we establish a criterion for the stability of the Kähler-Ricci flow (with perturbed complex structure) around a Kähler-Einstein metric with positive scalar curvature, under certain local stable condition on the dimension of holomorphic vector fields. In particular this gives a stability theorem for the existence of Kähler-Einstein metrics on a Kähler manifold with possibly nontrivial holomorphic vector fields.

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