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We study the convergence of the K\"ahler-Ricci flow on a compact K\"ahlermanifold $(M,J)$ with positive first Chern class $c_1(M;J)$ and vanished Futakiinvariant on $\pi c_1(M;J)$. As the application we establish a criterion forthe stability of the K\"ahler-Ricci flow (with perturbed complex structure)around a K\"ahler-Einstein metric with positive scalar curvature, under certainlocal stable condition on the dimension of holomorphic vector fields. Inparticular this gives a stability theorem for the existence ofK\"ahler-Einstein metrics on a K\"ahler manifold with possibly nontrivialholomorphic vector fields.

K\"ahler Ricci flow with vanished Futaki invariant