The set of destabilizing curves for deformed Hermitian Yang-Mills and Z-critical equations on surfaces

2022/10/12 by Sohaib Khalid, Zakarias Sjöström Dyrefelt, Khalid, Sohaib +1 · 2 citations
Mathematics · #14J60 #32Q26 #53C07 #53C55 #53E30 #Advanced Mathematical Physics Problems #Algebraic Geometry (math.AG) #Differential Geometry (math.DG) #FOS: Mathematics #Geometry and complex manifolds

paper · pdf · doi:10.48550/arxiv.2210.06374

openalex publication_date 2022/10/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We show that on any compact Kähler surface existence of solutions to the Z-critical equation can be characterized using a finite number of effective conditions, where the number of conditions is bounded above by the Picard number of the surface.This leads to a first PDE analogue of the locally finite wall-chamber decomposition in Bridgeland stability. As an application we characterize optimally destabilizing curves for Donaldson's J-equation and the deformed Hermitian Yang-Mills equation, prove a non-existence result for optimally destabilizing test configurations for uniform J-stability, and remark on improvements to convergence results for certain geometric flows.

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• The Critical LYZ equation in Kähler Geometry
• Polynomial stability conditions for vector bundles: Positivity, equivariance and blow-ups

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