$J$-equations and deformed Hermitian-Yang-Mills equations on holomorphic submersions
Abstract
In this paper, we prove that there exists a solution of the -equation on the total space of a holomorphic submersion if there exist solutions of the -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 -nef, then each fiber is -nef. In addition, if each fiber has a solution of the -equation, then the base is also -nef. Furthermore, we establish similar phenomena for the deformed Hermitian-Yang-Mills equation.
Keywords
Cite
@article{arxiv.2208.08576,
title = {$J$-equations and deformed Hermitian-Yang-Mills equations on holomorphic submersions},
author = {Rei Murakami},
journal= {arXiv preprint arXiv:2208.08576},
year = {2026}
}
Comments
Accepted 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