Hessian of the Ricci Calabi functional

The Ricci Calabi functional is a functional on the space of K\"ahler metrics of Fano manifolds. Its critical points are called generalized K\"ahler Einstein metrics. In this article, we show that the Hessian of the Ricci Calabi functional is non-negative at generalized K\"ahler Einstein metrics. As its application, we give another proof of a Matsushima's type decomposition theorem for holomorphic vector fields, which was originally proved by Mabuchi. We also discuss a relation to the inverse Monge-Amp\`ere flow developed recently by Collins-Hisamoto-Takahashi.