The Set of Destabilizing Curves for Deformed Hermitian Yang–Mills and Z-Critical Equations on Surfaces

Abstract

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.