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

Ramesh Mete

TL;DR

The paper addresses singularity formation in the deformed Hermitian Yang–Mills cotangent flow on the blow-up of $\mathbb{C}P^3$ at a point, using Calabi symmetry to reduce the problem to a one-dimensional ODE. It constructs an explicit unstable example where the flow develops a singularity along the exceptional divisor and shows that the limit satisfies a singular dHYM equation in the sense of DMS24, providing evidence for Conjecture 1.12 in this symmetry-reduced setting. The work also develops a framework of stability criteria, auxiliary deformations, and comparison principles to analyze long-time existence and convergence, distinguishing stable, semistable, and unstable regimes. The results illustrate infinite-time singularity formation along a divisor in dimension three and offer insight into how stability notions govern the asymptotic behavior of dHYM-type flows on higher-dimensional Kähler manifolds, with potential implications for non-pluripolar products and Bedford–Taylor-type limits.

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.

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

The paper addresses singularity formation in the deformed Hermitian Yang–Mills cotangent flow on the blow-up of at a point, using Calabi symmetry to reduce the problem to a one-dimensional ODE. It constructs an explicit unstable example where the flow develops a singularity along the exceptional divisor and shows that the limit satisfies a singular dHYM equation in the sense of DMS24, providing evidence for Conjecture 1.12 in this symmetry-reduced setting. The work also develops a framework of stability criteria, auxiliary deformations, and comparison principles to analyze long-time existence and convergence, distinguishing stable, semistable, and unstable regimes. The results illustrate infinite-time singularity formation along a divisor in dimension three and offer insight into how stability notions govern the asymptotic behavior of dHYM-type flows on higher-dimensional Kähler manifolds, with potential implications for non-pluripolar products and Bedford–Taylor-type limits.

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 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 in \cite{DMS24} on this three-dimensional manifold with symmetry.
Paper Structure (10 sections, 13 theorems, 115 equations)

This paper contains 10 sections, 13 theorems, 115 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Calabi symmetry and reduction
  4. Stability Inequalities
  5. Auxiliary family of equations
  6. Examples of two kinds of dHYM instability
  7. Proof of the main theorem
  8. Comparison and monotonicity along the flow
  9. A choice of initial function
  10. Proof of the convergence

Key Result

Theorem 1.2

Let $X=\mathrm{Bl}_{x_0}\mathbb{P}^3$ with two Kähler classes $\beta=3[H]-[E]$ and $\alpha=18[H]-3[E]$, and $\omega\in\beta$ be any Kähler form satisfying Calabi ansatz (see $\S$subsec:Calabi-symm-reduction). Let $\hat{\theta}\in(0,\pi)$ satisfying eq:basic cohomological condition, and $c:= \cot\hat

Theorems & Definitions (37)

  • Definition 1.1
  • Theorem 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • ...and 27 more