Stability and the deformed Hermitian-Yang-Mills equation

2020/04/09 by Tristan C. Collins, Yun Shi, Collins, Tristan C. +1 · 2 citations
Mathematics · Physics and Astronomy · #Algebraic Geometry (math.AG) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Nonlinear Waves and Solitons

paper · pdf · doi:10.48550/arxiv.2004.04831

openalex publication_date 2020/04/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We survey some recent progress on the deformed Hermitian-Yang-Mills (dHYM) equation. We discuss the role of geometric invariant theory (GIT) in approaching the solvability of the dHYM equation, following work of the first author and S.-T. Yau. We compare the GIT picture with the conjectural picture for dHYM involving Bridgeland stability. In particular, following Arcara-Miles, we show that on the blow-up of ℙ² any line bundle admitting a solution of the deformed Hermitian-Yang-Mills equation is Bridgeland stable, but not conversely. Finally, we survey some recent progress on heat flows associated to the dHYM equation.

Citations

• The Self-Duality Equations on a Riemann Surface
• (1,1) forms with specified Lagrangian phase: A priori estimates and\n algebraic obstructions
• The Dirichlet problem for nonlinear second order elliptic equations, III: Functions of the eigenvalues of the Hessian
• Iterated logarithms and gradient flows
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Cited by

• Geometric invariants and the Monge-Ampere equation in Kähler geometry
• Special Lagrangian sections and stability conditions on threefolds

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