Convergence of the $J$-flow on toric manifolds

TL;DR

This paper proves that on toric manifolds, the convergence of the J-flow is independent of the background metric and provides a numerical criterion for convergence, confirming a conjecture and extending results on inverse sigma_k equations.

Contribution

It offers a numerical characterization for J-flow convergence on toric manifolds, verifying a conjecture and strengthening existing inverse sigma_k equations results.

Findings

J-flow convergence is metric-independent on Kahler manifolds.

A numerical criterion determines J-flow convergence on toric manifolds.

Strengthened results on inverse sigma_k equations on Kahler manifolds.

Abstract

We show that on a Kahler manifold whether the J-flow converges or not is independent of the chosen background metric in its Kahler class. On toric manifolds we give a numerical characterization of when the J-flow converges, verifying a conjecture of Lejmi and the second author in this case. We also strengthen existing results on more general inverse $\sigma_{k}$ equations on Kahler manifolds.