A Kollár-type vanishing theorem for k-positive vector bundles

Given a proper holomorphic surjective morphism f : X → Y {f:X\rightarrow Y} between compact Kähler manifolds, and a Nakano semipositive holomorphic vector bundle E on X, we prove Kollár-type vanishing theorems on cohomologies with coefficients in R q ⁢ f ∗ ⁢ ( ω X ⁢ ( E ) ) ⊗ F {R^{q}f_{\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 {L^{2}} -Dolbeault resolution of the higher direct image sheaf R q ⁢ f ∗ ⁢ ( ω X ⁢ ( E ) ) {R^{q}f_{\ast}(\omega_{X}(E))} , which is of interest in itself.