An $L^1$ bound for Ricci curvature on Kähler manifolds

We prove a uniform $L^1$ bound for Ricci curvature on noncollapsed Kähler manifolds with Ricci curvature bounded below. The proof combines the codimension two neck decomposition and entropy estimates with local Kähler geometry on regular balls. The key step is a linear control of the Ricci integral on a regular ball by the entropy pinching, obtained from a decomposition of the scalar-curvature integral that exploits the Kähler structure.