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Numerical criteria on the complex Hessian quotient equations with the Calabi symmetry

Rei Murakami

TL;DR

This work establishes a Calabi-symmetry–driven numerical criterion that guarantees solvability of the complex Hessian quotient equation on a compact Kähler manifold, in line with Székelyhidi's conjecture. By employing the Calabi ansatz, the PDE is reduced to a first-order nonlinear ODE, and a boundary-value problem with $y(0)=0$ and $y(b)=q$ is solved under the numerical condition, ensuring the eigenvalue data remain in the admissible cone $\Gamma_k$. The paper also proves the existence of strictly $\alpha$-$(\omega,k)$-subharmonic representatives in the semiample case, using an Iitaka-fibration–based perturbation scheme, and discusses the relation to $(n-k)$-positivity in Andreotti-Grauert theory via an appendix. Together, these results advance understanding of partial positivity in cohomology classes for complex Hessian-type equations and connect to broader positivity criteria in Kähler geometry.

Abstract

Assuming Calabi symmetry, we prove that a numerical condition ensures the solvability of the complex Hessian quotient equation, as conjectured by Székelyhidi. We also propose a conjecture on the existence of a $k$-subharmonic representative in a given cohomology class and confirm it under the assumption of Calabi symmetry or when the class is semiample.

Numerical criteria on the complex Hessian quotient equations with the Calabi symmetry

TL;DR

This work establishes a Calabi-symmetry–driven numerical criterion that guarantees solvability of the complex Hessian quotient equation on a compact Kähler manifold, in line with Székelyhidi's conjecture. By employing the Calabi ansatz, the PDE is reduced to a first-order nonlinear ODE, and a boundary-value problem with and is solved under the numerical condition, ensuring the eigenvalue data remain in the admissible cone . The paper also proves the existence of strictly --subharmonic representatives in the semiample case, using an Iitaka-fibration–based perturbation scheme, and discusses the relation to -positivity in Andreotti-Grauert theory via an appendix. Together, these results advance understanding of partial positivity in cohomology classes for complex Hessian-type equations and connect to broader positivity criteria in Kähler geometry.

Abstract

Assuming Calabi symmetry, we prove that a numerical condition ensures the solvability of the complex Hessian quotient equation, as conjectured by Székelyhidi. We also propose a conjecture on the existence of a -subharmonic representative in a given cohomology class and confirm it under the assumption of Calabi symmetry or when the class is semiample.

Paper Structure

This paper contains 7 sections, 13 theorems, 52 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of Theorem \ref{['thm: main']}
  4. Set-up
  5. The solution of the boundary value problem
  6. Proof of Theorem \ref{['thm: semiample']}
  7. Relation to $q$-positivity in the sense of Andreotti-Grauert

Key Result

Theorem 1.2

Let $(X,\omega)$ be an $n$-dimensional compact Kähler manifold and $\alpha$ a closed real $(1,1)$-form on $X$. Assume that there exists a strictly $\alpha$-$(\omega,k)$-subharmonic function (see the definition below).

Theorems & Definitions (27)

  • Theorem 1.2: Sze
  • Definition 1.3: $\alpha$-$(\omega,k)$-subharmonicity
  • Definition 1.4
  • Conjecture 1.5
  • Remark 1.6
  • Theorem 1.8
  • Theorem 1.9
  • Definition 2.1
  • Proposition 2.2: Spr
  • Proposition 2.3: The Newton inequality, Spr
  • ...and 17 more