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

TL;DR

This paper proves a numerical criterion for solving complex Hessian quotient equations under Calabi symmetry and explores related conjectures on $k$-subharmonic representatives in cohomology classes.

Contribution

It establishes a solvability criterion for complex Hessian quotient equations assuming Calabi symmetry and proposes a conjecture on $k$-subharmonic representatives, confirming it in specific cases.

Findings

Numerical condition guarantees solvability under Calabi symmetry

Conjecture on $k$-subharmonic representatives proposed and partially confirmed

Results support Székelyhidi's conjecture in symmetric or semiample cases

Abstract

Assuming Calabi symmetry, we prove that a numerical condition ensures the solvability of the complex Hessian quotient equation, as conjectured by Sz\'ekelyhidi. 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.