An $L^2$ Dolbeault lemma on higher direct images and its application

Given a proper holomorphic surjective morphism $f:X\rightarrow Y$ from a compact K\"ahler manifold to a compact K\"ahler manifold, and a Nakano semipositive holomorphic vector bundle $E$ on $X$, we prove Koll\'ar type vanishing theorems on cohomologies with coefficients in $R^qf_\ast(\omega_X(E))\otimes F$, where $F$ is a $k$-positive vector bundle on $Y$. The main inputs in the proof are the deep results on the Nakano semipositivity of the higher direct images due to Berndtsson and Mourougane-Takayama, and an $L^2$-Dolbeault resolution of the higher direct image sheaf $R^qf_\ast(\omega_X(E))$, which is of interest in itself.