Compactness of Kahler-Ricci solitons on Fano manifolds

In this short paper, we improve the result of Phong-Song-Sturm on degeneration of Fano K"ahler-Ricci solitons by removing the assumption on the uniform bound of the Futaki invariant. Let $\mathcal{K}\mathcal{R}(n)$ be the space of K"ahler-Ricci solitons on $n$-dimensional Fano manifolds. We show that after passing to a subsequence, any sequence in $\mathcal{K}\mathcal{R}(n)$ converge in the Gromov-Hausdorff topology to a K"ahler-Ricci soliton on an $n$-dimensional $\mathbb{Q}$-Fano variety with log terminal singularities.