Gradient steady Kahler Ricci solitons with non-negative Ricci curvature and integrable scalar curvature

We study the non Ricci flat gradient steady K"{a}hler Ricci soliton with non-negative Ricci curvature and weak integrability condition of the scalar curvature $S$, namely ${\underset{}{\lim }}_{r\to \mathrm{\infty }}{r}^{-1}{\int }_{B_r}S=0$, and show that it is a quotient of $\mathrm{\Sigma }\times {\mathbb{C}}^{n-1-k}\times N^k$, where $\mathrm{\Sigma }$ and $N$ denote the Hamilton’s cigar soliton and some compact K"{a}hler Ricci flat manifold respectively. As an application, we prove that any non Ricci flat gradient steady K"{a}hler Ricci soliton with $Ric\ge 0$, together with subquadratic volume growth or ${\mathrm{lim\hspace{0.17em}sup}}_{r\to \mathrm{\infty }}rS<1$ must have universal covering space isometric to $\Sigma\times $\mathbb{C}$^{n-1-k}\times N^k$.