Uniform K-stability of polarized spherical varieties

Delcroix, Thibaut

doi:10.48550/arxiv.2009.06463

Cited by 9 publications

(20 citation statements)

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“…The equivariant test configurations for homogeneous toric bundles are encoded by rational piecewise-linear convex functions on ∆. See [31] for toric fibrations and [13] for spherical varieties. Repeating the argument of Sziékelyhidi, one can check that any convex function on ∆ gives rise to a filtration of the homogeneous coordinate ring.…”

Section: Homogeneous Toric Bundlesmentioning

confidence: 99%

“…Remark 1.5. In [13], Delcroix expressed K-stability of polarized spherical varieties in terms of combinatorial data and provided a combinatorial sufficient condition of Guniform K-stability. The appendix of [13] by Yuji Odaka showed that for non-singular spherical varieties, G-uniform K-stability is equivalent to existence of cscK metrics.…”

Section: Introductionmentioning

confidence: 99%

“…In [13], Delcroix expressed K-stability of polarized spherical varieties in terms of combinatorial data and provided a combinatorial sufficient condition of Guniform K-stability. The appendix of [13] by Yuji Odaka showed that for non-singular spherical varieties, G-uniform K-stability is equivalent to existence of cscK metrics. Jubert (see [25]) showed the equivalence between the existence of extremal Kähler metrics on the total space of semi-simple principal toric fibrations and weighted uniform K-stability of the corresponding Delzant polytopes.…”

Section: Introductionmentioning

confidence: 99%

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Extremal metrics on toric manifolds and homogeneous toric bundles

Li,

Lian,

Sheng

2021

Preprint

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Section: Homogeneous Toric Bundlesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Extremal metrics on toric manifolds and homogeneous toric bundles

Li,

Lian,

Sheng

2021

Preprint

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“…The following criterions for G-uniform K-stability of L follow from [Del20c]. Note that we switch here from concave to convex functions to simplify notations.…”

Section: Finally For Any Continuous Function G Onmentioning

confidence: 99%

“…The content of the note is as follows. In Section 2 we explain how projective cohomogeneity one manifolds coincide with (non-singular) rank one spherical varieties, briefly recall their Date: 2020. classification, then recall some of the results in [Del20c] for the special case of rank one spherical varieties. Section 3 is devoted to the proof of our main theorem, and of the corresponding Kstability statement which holds for singular varieties as well.…”

Section: Introductionmentioning

confidence: 96%

The Yau-Tian-Donaldson conjecture for cohomogeneity one manifolds

Delcroix¹

2020

Preprint

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We prove the Yau-Tian-Donaldson conjecture for cohomogeneity one manifolds, that is, for projective manifolds equipped with a holomorphic action of a compact Lie group with at least one real hypersurface orbit. Contrary to what seems to be a popular belief, such manifolds do not admit extremal Kähler metrics in all Kähler classes in general. More generally, we prove that for rank one polarized spherical varieties, G-uniform K-stability is equivalent to K-stability with respect to special G-equivariant test configurations. This is furthermore encoded by a single combinatorial condition, checkable in practice. We illustrate on examples and answer along the way a question of Kanemitsu.

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Valuative stability of polarised varieties

Dervan¹,

Legendre²

2020

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Fujita and Li have given a characterisation of K-stability of a Fano variety in terms of quantities associated to valuations, which has been essential to all recent progress in the area. We introduce a notion of valuative stability for arbitrary polarised varieties, and show that it is equivalent to Kstability with respect to test configurations with integral central fibre. The numerical invariant governing valuative stability is modelled on Fujita's βinvariant, but includes a term involving the derivative of the volume. We give several examples of valuatively stable and unstable varieties, including the toric case. We also discuss the role that the δ-invariant plays in the study of valuative stability and K-stability of polarised varieties.

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Uniform K-stability of polarized spherical varieties

Cited by 9 publications

References 0 publications

Extremal metrics on toric manifolds and homogeneous toric bundles

Extremal metrics on toric manifolds and homogeneous toric bundles

The Yau-Tian-Donaldson conjecture for cohomogeneity one manifolds

Valuative stability of polarised varieties

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