A Numerical Criterion for the 2-Hessian Equation on Compact Kähler Manifolds

Jixiang Fu; Dekai Zhang; Ziyi Zhang

A Numerical Criterion for the 2-Hessian Equation on Compact Kähler Manifolds

Differential Geometry 2026-07-13 v1

Authors: Jixiang Fu , Dekai Zhang , Ziyi Zhang

Abstract

We show that a Nakai--Moishezon-type criterion associated with the complex $2$ -Hessian equation produces a Gauduchon class. In complex dimension three, this numerical criterion is equivalent to the existence of a smooth $2$ -admissible representative and hence to the solvability of the $2$ -Hessian equation. As consequences of these results, we prove the corresponding conjectures of Murakami for the complex Hessian equation and of Sz\'ekelyhidi for the Hessian quotient equation in dimension three. We also establish a boundary version of the above results.

Keywords

2-Hessian equation Kähler manifolds Nakai-Moishezon criterion Gauduchon class admissible metrics Hessian quotient equation

Cite

@article{arxiv.2607.11296,
  title  = {A Numerical Criterion for the 2-Hessian Equation on Compact Kähler Manifolds},
  author = {Jixiang Fu and Dekai Zhang and Ziyi Zhang},
  journal= {arXiv preprint arXiv:2607.11296},
  year   = {2026}
}