Positivity in convergence of the inverse $\sigma_{n-1}$-flow

TL;DR

This paper investigates positivity conditions related to the inverse $\sigma_{n-1}$-flow, providing partial verification of a conjecture on the flow's convergence and solvability in certain geometric contexts.

Contribution

It offers new positivity results for the inverse $\sigma_{n-1}$-flow and verifies the conjecture in specific cases like $(n-1, n-1)$ classes and 3-folds.

Findings

Positivity established for $(n-1, n-1)$ cohomology classes.

Partial verification of the conjecture for 3-folds.

Advances understanding of conditions for flow convergence.

Abstract

We study positivity in the conjecture proposed by Lejmi and Sz\'{e}kelyhidi on finding effective necessary and sufficient conditions for solvability of the inverse $\sigma_k$ equation, or equivalently, for convergence of the inverse $\sigma_k$-flow. In particular, for the inverse $\sigma_{n-1}$-flow we partially verify their conjecture by obtaining the desired positivity for $(n-1, n-1)$ cohomology classes. As an application, we also partially verify their conjecture for 3-folds.