A note on modified $J$-Flow with the Calabi Ansatz

TL;DR

This paper investigates the behavior of the modified J-flow under Calabi symmetry, revealing conditions under which the flow converges or blows up, and linking convergence to topological constants and Hamiltonian minima.

Contribution

It demonstrates that the modified J-flow can blow up when certain intersection numbers are non-positive, and characterizes convergence behavior based on topological data and symmetry.

Findings

Flow blows up when intersection numbers are non-positive.

Flow converges to solutions away from singularities.

Convergence depends on topological constants and Hamiltonian minima.

Abstract

We study the modified $J$-flow introduced in [15], particularly the singularities of the flow using the Calabi symmetry. In [20], on toric manifolds the convergence of modified $J$-flow to the smooth solution was proven under the assumption of positivity of certain intersection numbers. In the case of the Calabi ansatz, we show that if some of those intersection numbers are not positive, then the modified $J$-flow blows up along some variety, and away from the variety we prove the convergence to the solution. As in [10], we also prove that the convergence behavior of the modified $J$-flow with Calabi symmetry depends on the topological constants $c$ and the minimum of the Hamiltonian function.