Psh variation of singular Kähler-Einstein metrics and Invariance of plurigenera for smooth Kähler families

Abstract

Let $p:X\to \Delta$ be a proper holomorphic submersion whose total space $X$ is a Kähler manifold. We prove that ${\dim }_{\mathbb{C}}{H}^0(X_t,m{K}_{X_t})$ is independent of $t$ for every integer $m\ge 1$. This was conjectured by Y.-T. Siu in 2002.