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

TL;DR

This paper characterizes the existence of solutions to Z-critical equations on Kähler surfaces using finitely many conditions, linking stability concepts with geometric PDEs and providing insights into destabilizing curves and flow convergence.

Contribution

It introduces a finite condition criterion for Z-critical solutions on surfaces, connecting stability theory with PDE analysis and destabilization phenomena.

Findings

Finite conditions characterize Z-critical solutions.

Identifies destabilizing curves for specific geometric equations.

Provides non-existence results for certain test configurations.

Abstract

We show that on any compact K\"ahler 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.