Perelman's W-functional and stability of K？hler-Ricci flow

In this note we study the second variation of Perelman's W-functional on the space of K\"ahler metrics at a K\"ahler-Ricci soliton (in particular, at a K\"ahler-Einstein metric) and its applications. We prove that Perelman's W-functional is stable in the space of K\"ahler metrics with K\"ahler classes cohomologous to $2\pi c_1(M)$ (complex structures on $M$ may vary) in the sense of variations. As an application, a stability theorem about K\"ahler-Ricci flow near a K\"ahler-Einstein metric is proved.