A Rigidity Theorem for the deformed Hermitian-Yang-Mills equation

Han, Xiaohui; Jin, Xuecheng

doi:10.1007/s00526-020-01880-9

Cited by 7 publications

(2 citation statements)

References 23 publications

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“…Existence criteria have been studied in a large body of work (see, for example, [8,50,40,14]) culminating in [6] and [16,51] with a characterization of existence in terms of uniform J-stability, mirroring the Yau-Tian-Donaldson conjecture, and relating stability to analytic conditions tested on subvarieties, as conjectured by Lejmi-Székelyhidi [40]. The dHYM equation has been studied in its own right in a large body of recent work [9,11,10,12,13,15,31,32,33,36,29,37,39,43], and the Z-critical equation was recently introduced by Dervan-McCarthy-Sektnan [22,24,42] as a general framework extending and encompassing several geometric PDE of interest in Kähler geometry -including the dHYM equation -and with the aim of highlighting connections with algebro-geometric stability conditions in the sense of Bridgeland [4].…”

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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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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“…Recently, the dHYM metrics have been studied actively(e.g. [2], [4], [5], [7], [8], [9], [14], [15], [16] etc).…”

Section: Introductionmentioning

confidence: 99%

Stability of line bundle mean curvature flow

Han¹,

Jin²

2020

Preprint

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Let (X, ω) be a compact Kähler manifold of complex dimension n and (L, h) be a holomorphic line bundle over X. The line bundle mean curvature flow was introduced by Jacob-Yau in order to find deformed Hermitian-Yang-Mills metrics on L. In this paper, we consider the stability of the line bundle mean curvature flow. Suppose there exists a deformed Hermitian Yang-Mills metric ĥ on L. We prove that the line bundle mean curvature flow converges to ĥ exponentially in C ∞ sense as long as the initial metric is close to ĥ in C 2 -norm.

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A numerical criterion for generalised Monge-Ampère equations on projective manifolds

Datar

Pingali

2021

Geom. Funct. Anal.

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We prove that generalised Monge-Ampère equations (a family of equations which includes the inverse Hessian equations like the J-equation, as well as the Monge-Ampère equation) on projective manifolds have smooth solutions if certain intersection numbers are positive. As corollaries of our work, we improve a result of Chen (albeit in the projective case) on the existence of solutions to the J-equation, and prove a conjecture of Székelyhidi in the projective case on the solvability of certain inverse Hessian equations. The key new ingredient in improving Chen's result is a degenerate concentration of mass result. We also prove an equivariant version of our results, albeit under the assumption of uniform positivity. In particular, we can recover existing results on manifolds with large symmetry such as projective toric manifolds.

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A Rigidity Theorem for the deformed Hermitian-Yang-Mills equation

Cited by 7 publications

References 23 publications

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

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

Stability of line bundle mean curvature flow

A numerical criterion for generalised Monge-Ampère equations on projective manifolds

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