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

TL;DR

This paper demonstrates the existence of solutions to the $J$-equation and deformed Hermitian-Yang-Mills equations on holomorphic submersions by relating solutions on fibers, the base, and the total space using adiabatic limit techniques.

Contribution

It establishes conditions under which solutions on fibers and base imply solutions on the total space, advancing understanding of these equations on complex fibrations.

Findings

Solutions on fibers and base imply solutions on the total space.

If the total space is $J$-nef, then fibers are $J$-nef.

Solutions on fibers imply the base is $J$-nef.

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.