Singularity formation in co-dimension one of the dHYM cotangent flow on blow up of $\mathbb{C}\mathbb{P}^{3}$ at a point

TL;DR

This paper investigates singularity formation in the deformed Hermitian Yang-Mills (dHYM) cotangent flow on a blow-up of complex projective space, providing explicit examples and evidence for related conjectures.

Contribution

It offers the first explicit example of singularity development in the dHYM flow on a three-dimensional manifold with symmetry, using Calabi ansatz.

Findings

Flow develops a singularity along the exceptional divisor.

Limit satisfies a singular dHYM equation in the sense of prior work.

Provides evidence supporting a conjecture in the field.

Abstract

The existence and uniqueness of canonical singular solutions of the J-equation and the deformed Hermitian Yang Mills (dHYM) equation was proved in \cite{DMS24} on compact K\"{a}hler surfaces. In this paper, we study the singularity formation of the dHYM cotangent flow on the one-point blow up of $\mathbb{C}\mathbb{P}^3$ using Calabi ansatz. In particular, we provide an explicit example where the flow develops a singularity along the exceptional divisor. Moreover, the limit satisfies corresponding singular dHYM equation in the sense of \cite{DMS24} and provides some evidence for Conjecture $1.12$ in \cite{DMS24} on this three-dimensional manifold with symmetry.