The closures of test configurations and algebraic singularity types

Darvas, Tamás; Xia, Mingchen

doi:10.48550/arxiv.2003.04818

Cited by 7 publications

(21 citation statements)

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“…which means that the point associated to E j in the twisted test configuration is given by v j,−ξ . So we use the formula (35) for Monge-Ampère measure to get:…”

Section: G-uniform K-stabilitymentioning

confidence: 99%

“…See the recent preprint in [35] for more equivalent characterizations of maximal geodesic rays. The proof of Theorem 2.34 hinges on the following important construction by Berman-Boucksom-Jonsson, which in particular shows that any φ ∈ E 1,NA (L) can be approximated by a decreasing sequence {φ m } ⊂ H NA .…”

Section: Maximal Geodesic Rays and Finite Energy Non-archimedean Metricsmentioning

confidence: 99%

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Geodesic rays and stability in the cscK problem

Li¹

2020

Preprint

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We prove that any finite energy geodesic ray with a finite Mabuchi slope is maximal in the sense of Berman-Boucksom-Jonsson, and reduce the proof of the uniform Yau-Tian-Donaldson conjecture for constant scalar curvature Kähler metrics to Boucksom-Jonsson's regularization conjecture about the convergence of non-Archimedean entropy functional. As further applications, we show that a uniform K-stability condition for model filtrations and the J K X -stability are both sufficient conditions for the existence of cscK metrics. The first condition is also conjectured to be necessary. Our arguments also produce a different proof of the toric uniform version of YTD conjecture for all polarized toric manifolds. Another result proved here is that the Mabuchi slope of a geodesic ray associated to a test configuration is equal to the non-Archimedean Mabuchi invariant.Conjecture 1.8 (Regularization Conjecture,[20]). For any φ ∈ E 1,NA (L), there exists a sequence {φ m } ⊂ H NA (L) converging to φ in the strong topology such that:Indeed, the results proved here can show that the conjecture 1.8 implies both Conjecture 1.5 and Conjecture 1.6, and hence also the uniform version of YTD conjecture (see Lemma 6.8 and Proposition 6.7). Although Conjecture 1.8 is not known in general yet, we can nevertheless use Boucksom-Jonsson's non-Archimedean approach to K-stability ([19, 20]) and Boucksom-Favre-Jonsson's foundational work on non-Archimedean Monge-Ampère equations ([14, 19, 20]) to get an existence result involving K-stability for model filtrations.To state a general existence statement, let G be a reductive complex Lie group in Aut(X, L) 0 with a maximal compact subgroup K such that K C = G. Let T = ((S 1 ) r ) C be the identity component of the center of

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“…which means that the point associated to E j in the twisted test configuration is given by v j,−ξ . So we use the formula (35) for Monge-Ampère measure to get:…”

Section: G-uniform K-stabilitymentioning

confidence: 99%

Section: Maximal Geodesic Rays and Finite Energy Non-archimedean Metricsmentioning

confidence: 99%

Geodesic rays and stability in the cscK problem

Li¹

2020

Preprint

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“…The part for J follows from the strategy introduced in [RWN14] and further developed in [DX20]. See the proof of Theorem 6.7 for details.…”

Section: Strategy Of the Proofmentioning

confidence: 99%

“…In [Li20], Li showed that Ent NA (φ) is dominated by the slope at infinity of the usual entropy functional Ent along ℓ, where φ is the non-Archimedean potential induced by ℓ. In [DX20], we have expressed the non-Archimedean Monge-Ampère energy in terms of the test curves. Since the non-Archimedean Monge-Ampère energy is nothing but the primitive function of the Chambert-Loir measure MA(φ), we get a fortiori a good understanding of MA(φ).…”

Section: Strategy Of the Proofmentioning

confidence: 99%

Pluripotential-theoretic stability thresholds

Xia¹

2020

Preprint

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Given a compact polarized manifold (X, L), we introduce two new stability thresholds in terms of singularity types of global quasi-plurisubharmonic functions on X. We prove that in the Fano setting, the new invariants can effectively detect the K-stability of X. We study some functionals of geodesic rays in the space of Kähler potentials by means of the corresponding test curves. In particular, we introduce a new entropy functional of quasi-plurisubharmonic functions and relate the radial entropy functional to this new entropy functional.

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“…Proof. The argument uses the formalism of Legendre transforms for geodesic rays going back to [RWN14], further developed in [DX20]. The first estimate is trivial.…”

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confidence: 99%