K-stability for Kähler manifolds

Dervan, Ruadhaí; Ross, Julius

doi:10.4310/mrl.2017.v24.n3.a5

Cited by 55 publications

(114 citation statements)

References 56 publications

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“…Firstly, we will relate the norms of test configurations and their corresponding geodesics. Using this, we will also be able to characterise the trivial Kähler test configurations, clarifying the definition of K-stability given in [24] (by the pathological examples of Li-Xu [34], it is a rather subtle problem to understand what it means for a test configuration to be trivial, even in the projective case). The minimum norm [22] is also called the "non-Archimedean J-functional" [12].…”

Section: Here T Denotes the Set Of Test Configurations For (X [ω]) mentioning

confidence: 99%

“…Although this article is essentially a sequel to [24,43], where Theorem 1.1 was proven, the techniques used are very different. In [24,43] the main theme was to differentiate energy functionals on the space of Kähler metrics along certain paths induced by test configurations.…”

Section: Comparison With Other Workmentioning

confidence: 99%

“…We now introduce a notion of relative K-stability for Kähler manifolds, generalising the notion of K-stability defined in [24,43]. We refer to [21,24] for the background on Kähler analytic spaces needed.…”

Section: Relative K-stability For Kähler Manifoldsmentioning

confidence: 99%

“…Analogues of Theorem 1.4 can be proven using similar methods for twisted cscK metrics and the J-flow. Here one replaces the left hand side with S(ω) − α ω − c 1 p for twisted cscK metrics, and with α ω − c 2 p for the J-flow (where c 1 , c 2 are the appropriate topological constants and α is an auxiliary Kähler metric in an arbitrary Kähler class), and replaces the right hand side with the corresponding numerical invariants [22,31] (for the J-flow one should use the numerical invariants as formulated in [23,Section 4.2] rather than the original formulation in [31], as in [24,Section 6]). In the projective case, these results were proven in [22,31] for p = q = 2.…”

Section: Here T Denotes the Set Of Test Configurations For (X [ω]) mentioning

confidence: 99%

“…In [24,43] the main theme was to differentiate energy functionals on the space of Kähler metrics along certain paths induced by test configurations. To prove Theorems 1.2 and 1.4 we instead use a combination of the results obtained in [24,43] together with a delicate use Hamiltonian geometry to obtain precise information about how the invariants of a test configuration (such as the norm) change when one perturbs the test configuration.…”

Section: Comparison With Other Workmentioning

confidence: 99%

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Relative K-stability for Kähler manifolds

Dervan

2017

Math. Ann.

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We study the existence of extremal Kähler metrics on Kähler manifolds. After introducing a notion of relative K-stability for Kähler manifolds, we prove that Kähler manifolds admitting extremal Kähler metrics are relatively K-stable. Along the way, we prove a general L p lower bound on the Calabi functional involving test configurations and their associated numerical invariants, answering a question of Donaldson. When the Kähler manifold is projective, our definition of relative K-stability is stronger than the usual definition given by Székelyhidi. In particular our result strengthens the known results in the projective case (even for constant scalar curvature Kähler metrics), and rules out a well known counterexample to the "naïve" version of the Yau-Tian-Donaldson conjecture in this setting.

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Section: Here T Denotes the Set Of Test Configurations For (X [ω]) mentioning

confidence: 99%

Section: Comparison With Other Workmentioning

confidence: 99%

Section: Relative K-stability For Kähler Manifoldsmentioning

confidence: 99%

Section: Here T Denotes the Set Of Test Configurations For (X [ω]) mentioning

confidence: 99%

Section: Comparison With Other Workmentioning

confidence: 99%

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Relative K-stability for Kähler manifolds

Dervan

2017

Math. Ann.

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Quantization in Geometric Pluripotential Theory

Darvas

Rubinstein

2019

Comm Pure Appl Math

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The space of Kähler metrics, on the one hand, can be approximated by subspaces of algebraic metrics, while, on the other hand, it can also be enlarged to finite‐energy spaces arising in pluripotential theory. The latter spaces are realized as metric completions of Finsler structures on the space of Kähler metrics. The former spaces are the finite‐dimensional spaces of Fubini‐Study metrics of Kähler quantization. The goal of this article is to draw a connection between the two. We show that the Finsler structures on the space of Kähler potentials can be quantized. More precisely, given a Kähler manifold polarized by an ample line bundle we endow the space of Hermitian metrics on powers of that line bundle with Finsler structures and show that the resulting path length metric spaces recover the corresponding metric completions of the Finsler structures on the space of Kähler potentials. This has a number of applications, among them a new Lidskii‐type inequality on the space of Kähler metrics, a new approach to the rooftop envelopes and Pythagorean formulas of Kähler geometry, and approximation of finite‐energy potentials, as well as geodesic segments by the corresponding smooth algebraic objects. © 2019 Wiley Periodicals, Inc.

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Twisted Kähler–Einstein metrics in big classes

Darvas,

Zhang

2024

Comm Pure Appl Math

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We prove existence of twisted Kähler–Einstein metrics in big cohomology classes, using a divisorial stability condition. In particular, when is big, we obtain a uniform Yau–Tian–Donaldson (YTD) existence theorem for Kähler–Einstein (KE) metrics. To achieve this, we build up from scratch the theory of Fujita–Odaka type delta invariants in the transcendental big setting, using pluripotential theory. We do not use the K‐energy in our arguments, and our techniques provide a simple roadmap to prove YTD existence theorems for KE type metrics, that only needs convexity of the appropriate Ding energy. As an application, we give a simplified proof of Li–Tian–Wang's existence theorem in the log Fano setting.

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K-stability for Kähler manifolds

Cited by 55 publications

References 56 publications

Relative K-stability for Kähler manifolds

Relative K-stability for Kähler manifolds

Quantization in Geometric Pluripotential Theory

Twisted Kähler–Einstein metrics in big classes

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