Singularity Formation Along the Line Bundle Mean Curvature Flow

Abstract The line bundle mean curvature flow is a complex analogue of the mean curvature flow for Lagrangian graphs, with fixed points solving the deformed Hermitian–Yang–Mills equation. In this paper, we construct two distinct examples of singularities along the flow. First, we find a finite time singularity, ruling out long time existence of the flow in general. Next we show long time existence of the flow with a Calabi symmetry assumption on the blowup of $\mathbb P^{n}$, $n\geq 3$, if one assumes supercritical phase. Using this, we find an example where a singularity occurs at infinite time along the destabilizing subvariety in the semi-stable case.