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

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