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

Given a proper holomorphic surjective morphism <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>f</m:mi> <m:mo>:</m:mo> <m:mrow> <m:mi>X</m:mi> <m:mo>→</m:mo> <m:mi>Y</m:mi> </m:mrow> </m:mrow> </m:math> {f:X→ 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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:msup> <m:mi>R</m:mi> <m:mi>q</m:mi> </m:msup> <m:mo>⁢</m:mo> <m:msub> <m:mi>f</m:mi> <m:mo>∗</m:mo> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:msub> <m:mi>ω</m:mi> <m:mi>X</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>E</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>⊗</m:mo> <m:mi>F</m:mi> </m:mrow> </m:math> {Rqf_{}(ωX(E))⊗ 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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>L</m:mi> <m:mn>2</m:mn> </m:msup> </m:math> {L²} -Dolbeault resolution of the higher direct image sheaf <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msup> <m:mi>R</m:mi> <m:mi>q</m:mi> </m:msup> <m:mo>⁢</m:mo> <m:msub> <m:mi>f</m:mi> <m:mo>∗</m:mo> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:msub> <m:mi>ω</m:mi> <m:mi>X</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>E</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {Rqf_{}(ωX(E))} , which is of interest in itself.