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

Rei Murakami

TL;DR

The paper develops a robust framework for solving the $J$-equation and the deformed Hermitian-Yang-Mills equation on the total space of a holomorphic submersion by leveraging fiberwise solutions and adiabatic (large $k$) scaling. It provides two independent existence proofs in the $J$-setting—one via the $ rak{C}$-subsolution criterion and another by a quantitative inverse function theorem—and establishes parallel results for dHYM, including a supercritical-phase route via CJY subsolutions and a general adiabatic-limit approach that avoids the supercritical constraint. A converse analysis based on $J$-positivity/nefness links fiber and base solvability to total-space behavior, with analogous statements for dHYM. The work also supplies explicit approximate solutions and applies its abstract results to concrete geometries such as projective bundles and blowups, broadening canonical metric existence results in complex geometry.

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.

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

TL;DR

The paper develops a robust framework for solving the -equation and the deformed Hermitian-Yang-Mills equation on the total space of a holomorphic submersion by leveraging fiberwise solutions and adiabatic (large ) scaling. It provides two independent existence proofs in the -setting—one via the -subsolution criterion and another by a quantitative inverse function theorem—and establishes parallel results for dHYM, including a supercritical-phase route via CJY subsolutions and a general adiabatic-limit approach that avoids the supercritical constraint. A converse analysis based on -positivity/nefness links fiber and base solvability to total-space behavior, with analogous statements for dHYM. The work also supplies explicit approximate solutions and applies its abstract results to concrete geometries such as projective bundles and blowups, broadening canonical metric existence results in complex geometry.

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.
Paper Structure (14 sections, 33 theorems, 153 equations)

This paper contains 14 sections, 33 theorems, 153 equations.

Table of Contents

  1. Introduction
  2. $J$-equation
  3. Preliminaries
  4. Approximate solutions
  5. First proof of Theorem \ref{['thm:main']}
  6. Second proof of Theorem \ref{['thm:main']}
  7. Proof of Theorem \ref{['thm:converse']}
  8. Deformed Hermitian-Yang-Mills equations
  9. Preliminaries
  10. Approximate solutions
  11. First proof of Theorem \ref{['thm:dhym_main']} under the supercritical phase condition
  12. Proof of Theorem \ref{['thm:dhym_main']}
  13. A variant of Theorem \ref{['thm:converse']}
  14. Examples

Key Result

Theorem 1.3

Suppose that $(B, L)$ admits a twisted cscK metric $\omega_B \in c_1(L)$ and $\pi: (X,H) \rightarrow (B,L)$ admits an optimal symplectic connection $\omega_X \in c_1(H)$. Assume that both of the automorphism groups $\mathrm{Aut}(X,H)$ and $\mathrm{Aut}(q)$, where $q$ is the moduli map, are discrete.

Theorems & Definitions (69)

  • Theorem 1.3: Dervan-Sektnan
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.9
  • Definition 2.2
  • Remark 2.3
  • Lemma 2.4: Hashimoto2018
  • proof
  • Definition 2.6
  • Lemma 2.7
  • ...and 59 more