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

Abstract

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 $C^\infty$ 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.