$J$-equations and deformed Hermitian-Yang-Mills equations on holomorphic submersions

In this paper, we prove that there exists a solution of the $J$-equation on the total space of a holomorphic submersion if there exist solutions of the $J$-equation on the fibers and the base. The method is an adiabatic limit technique. We also partially prove the converse implication. More precisely, if the total space is $J$-nef, then each fiber is $J$-nef. In addition, if each fiber has a solution of the $J$-equation, then the base is also $J$-nef. Furthermore, we establish similar phenomena for the deformed Hermitian-Yang-Mills equation.