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

Abstract

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.

arXiv commentAccepted version. An alternative proof of Theorem 1.4 using the $\mathcal{C}$-subsolution has been added. A section on the dHYM equation has been added, and the title has been changed accordingly. A section on examples has also been added at the end of the paper. The proof of Lemma 2.21 (Lemma 3.10 in the previous version) has been replaced with a clearer one. Other changes are minor. 35 pages