Collapsing of Products Along the K\"ahler-Ricci Flow

Let $X=M\times E$ where $M$ is an $m$-dimensional K"ahler manifold with negative first Chern class and $E$ is an $n$-dimensional complex torus. We obtain $C^{\mathrm{\infty }}$ convergence of the normalized K"ahler-Ricci flow on $X$ to a K"ahler-Einstein metric on $M$. This strengthens a convergence result of Song-Weinkove and confirms their conjecture.