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

Abstract

We study positivity in the conjecture proposed by Lejmi and Székelyhidi 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.