A decomposition formula for J-stability and its applications

Hattori, Masumi

doi:10.48550/arxiv.2103.04603

Cited by 5 publications

(21 citation statements)

References 46 publications

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“…The definition of J H -stability is of similar form (cf., [44]). On the other hand, (X, ∆, L) is K-polystable if for any normal semiample test configuration (X , L),…”

Section: Notationmentioning

confidence: 99%

“…Next, we see that if the twisted canonical class is positive, then an algebraic fiber space is adiabatically K-stable under some conditions. Recall the following fact, Theorem 3.3 (cf., [46,Theorem 1.1], [79,Theorem 1.1], [80,Corollary 1.3], [44,Theorem 8.15]). Let (X, ∆) be a klt minimal pair i.e., K X + ∆ is nef.…”

Section: Notationmentioning

confidence: 99%

“…Note that deg κ C ≥ 0 if and only if the log-twisted pair (C, B, M, L) is general type or Calabi-Yau in Theorem B. In this case, the theorem is solved in [44,Theorem G].…”

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confidence: 99%

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On K-stability of Calabi-Yau fibrations

Hattori¹

2022

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We study K-stability of algebraic fiber spaces. In this paper, we prove that Calabi-Yau fibrations over curves are uniformly K-stable in an adiabatic sense if and only if the base curves are log-twisted K-stable, where log-twisted K-stability is a variant of K-stability. Moreover, we prove that there are cscK metrics on rational elliptic surfaces that have only reduced fibers. Contents 14 4.2. Klt-trivial fibrations and log-twisted stability 20 5. Applications to Rational Elliptic Surfaces 27 5.1. Log-twisted K-stability of the base curve 27 5.2. GIT-stability in the sense of Miranda I: Weierstrass fibrations 29 5.3. GIT-stability in the sense of Miranda II: Cubic pencils 32 5.4. Applications to Miranda's conjecture on Chow stability 35 Appendix A. Some results on log-twisted Fano pairs 38 A.1. Notation 38 A.2. Ding stability of log-twisted Fano pairs and δ-invariant 40 A.3. Log-twisted K-semistable Fano pair and destabilizing boundary 44 References 49

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“…The definition of J H -stability is of similar form (cf., [44]). On the other hand, (X, ∆, L) is K-polystable if for any normal semiample test configuration (X , L),…”

Section: Notationmentioning

confidence: 99%

Section: Notationmentioning

confidence: 99%

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On K-stability of Calabi-Yau fibrations

Hattori¹

2022

Preprint

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“…Sjöström Dyrefelt [36] and Song [37] generalized independently the result of Jian-Shi-Song to the case when X is a smooth minimal model (i.e., K X is nef) later. On the other hand, the author proved K-stability of klt minimal models in [22].…”

Section: Introductionmentioning

confidence: 99%

“…for 2 ≤ j ≤ r. Let B = B k be the irreducible decomposition and B 1 = f (X 1 ). Then we obtain as [22,Theorem 6.6]…”

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confidence: 99%

On fibration stability after Dervan-Sektnan and singularities

Hattori¹

2022

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We introduce f-stability, a modification of fibration stability of Dervan-Sektnan [12], and show that f-semistable fibrations have only semi log canonical singularities. Moreover, f-stability puts restrictions on semi log canonical centers on Fano fibrations. Contents 1. Introduction 1 Outline of this paper 4 Acknowledgements 4 2. Notation 4 3. Review of Basic Results on K-stability and singularities 9 4. Fibration stability after Dervan-Sektnan 11 5. Singularities of semistable algebraic fiber spaces 18 5.1. Normal case 18 5.2. Non-normal case 22 References 26

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

Khalid¹,

Dyrefelt²

2022

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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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A decomposition formula for J-stability and its applications

Cited by 5 publications

References 46 publications

On K-stability of Calabi-Yau fibrations

On K-stability of Calabi-Yau fibrations

On fibration stability after Dervan-Sektnan and singularities

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

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