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

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.

Accepted version. The title has been changed for standard terminology (from "$(k,\ell )$-Hessian'' to "Hessian quotient''). Minor revisions. 16 pages