Invariance of plurigenera for K\"ahler families with nef canonical bundles

Abstract

Let $p:X\rightarrow\Delta$ be a proper holomorphic submersion from a K\"ahler manifold $X$. We prove that, if $K_{X_t}$ is nef for every $t\in\Delta$, then $\dim_{\mathbb C}H^0(X_t,mK_{X_t})$ is independent of $t$ for every integer $m\geq1$. This gives a partial answer to Siu's conjecture on the invariance of plurigenera for K\"ahler families.