An effective weighted K-stability condition for polytopes and semisimple principal toric fibratons

Delcroix, Thibaut; Jubert, Simon

doi:10.48550/arxiv.2202.02996

Cited by 1 publication

(1 citation statement)

References 21 publications

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“…conditions and optimal destabilizers have been investigated in a variety of different situations, such as toric geometry and its generalizations or K-stability for Fano varieties (see e.g. [45,1,35,57,17,18,19,23,53]). The general picture is that even if one manages to characterize existence via algebro-geometric stability, such as K-stability, this typically involves testing an infinite number of conditions, and is therefore in general still far from being testable in practice.…”

Section: Introductionmentioning

confidence: 99%

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

Khalid¹,

Dyrefelt²

2022

Preprint

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We show that on any compact Kähler surface obstructions to the existence of solutions to the J-equation, deformed Hermitian-Yang-Mills equation, and Z-critical equation can each be determined using a finite number of effective conditions, where the number of conditions needed is bounded above by the Picard number of the surface. The novel technique is the use of Zariski decomposition, which produces a finite set of 'test curves' uniform across compact sets of initial data. This shows that a finite number of polynomials locally cut out the set of classes for which the respective equations are solvable, giving a first analogue on the PDE side of the locally finite wall-and-chamber decomposition central in the theory of Bridgeland stability. We moreover deduce that the boundary of the set of classes admitting a solution to the dHYM and Z-critical equations form real algebraic sets of codimension one.As an application we characterize optimally destabilizing curves for the J-equation and dHYM equation, and prove a non-existence result for optimally destabilizing test configurations for uniform J-stability. In connection with this we show that on a large class of surfaces one can find polarizations that are J-stable but not uniformly J-stable. We finally remark on an improvement to convergence results for the J-flow, line bundle mean curvature flow, and dHYM flow.

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