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Given a proper holomorphic surjective morphism $f:X\rightarrow Y$ from acompact K\"ahler manifold to a compact K\"ahler manifold, and a Nakanosemipositive holomorphic vector bundle $E$ on $X$, we prove Koll\'ar typevanishing 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 Nakanosemipositivity of the higher direct images due to Berndtsson andMourougane-Takayama, and an $L^2$-Dolbeault resolution of the higher directimage sheaf $R^qf_\ast(\omega_X(E))$, which is of interest in itself.

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