  Let $(M,J_0)$ be a Fano manifold which admits a K\"ahler-Ricci soliton, we
analyze the behavior of the K\"ahler-Ricci flow near this soliton as we deform
the complex structure $J_0$. First, we will establish an inequality of
Lojasiewicz's type for Perelman's entropy along the K\"ahler-Ricci flow. Then
we prove the convergence of K\"ahler-Ricci flow when the complex structure
associated to the initial value lies in the kernel $Z$ or negative part of the
second variation operator of Perelman's entropy. As applications, we solve the
Yau-Tian-Donaldson conjecture for the existence of K\"ahler-Ricci solitons in
the moduli space of complex structures near $J_0$, and we show that the kernel
$Z$ corresponds to the local moduli space of Fano manifolds which are modified
$K$-semistable. We also prove an uniqueness theorem for K\"ahler-Ricci
solitons.
