Invariance of plurigenera for Kähler families with nef canonical bundles

Let $p:X\to \mathrm{\Delta }$ be a proper holomorphic submersion from a Kähler manifold $X$. We prove that, if $K_{X_t}$ is nef for every $t\in \mathrm{\Delta }$, then ${\dim }_{\mathbb{C}}{H}^0(X_t,m{K}_{X_t})$ is independent of $t$ for every integer $m\ge 1$. This gives a partial answer to Siu’s conjecture on the invariance of plurigenera for Kähler families.