$J$-equations on holomorphic submersions

Abstract: In this paper, we prove that there exists a solution of the $J$-equation on a total space of a holomorphic submersion if there exist solutions of the $J$-equation on fibres and a base. The method is an adiabatic limit technique. We also partially prove the converse implication. More precisely, if a total space is $J$-nef, then each fibre is $J$-nef. In addition, if each fibre has a solution of the $J$-equation, then a base is also $J$-nef.