Psh variation of singular K\"ahler-Einstein metrics and Invariance of plurigenera for smooth K\"ahler families

Abstract

Let $p:X\rightarrow\Delta$ be a proper holomorphic submersion whose total space $X$ is a K\"ahler manifold. We prove that $\dim_{\mathbb C}H^0(X_t,mK_{X_t})$ is independent of $t$ for every integer $m\geq1$. This was conjectured by Y.-T. Siu in 2002.