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

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 Kähler 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.

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