$J$-equations on holomorphic submersions

Abstract

In this paper, we prove that there exists a solution of the Jequation 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.