Invariance of plurigenera for Kähler families with nef canonical bundles

Let $p:X\rightarrowΔ$ be a proper holomorphic submersion from a Kähler manifold $X$. We prove that, if $K_{X_t}$ is nef for every $t\inΔ$, 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 p