Weighted K-stability of $\mathbb Q$-Fano spherical varieties

Feng Wang; Yan Li; Zhenye Li

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Weighted K-stability of $\mathbb Q$-Fano spherical varieties

Yan Li, ZhenYe Li, Feng Wang

Differential Geometry · Submitted 4 Aug 2022

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim Finally we show the equivalence of different notations of stability and gives a stability criterion on $\mathbb Q$-Fano spherical varieties, which is also a criterion of existence of K\"ahler-Ricci $g$-solitons.

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arXiv listing abstract v2. Results and the main claim are underlined; show every role.

Let $G$ be a connected, complex reductive Lie group and $X$ a $\mathbb Q$-Fano $G$-spherical variety. In this paper we compute the weighed non-Archimedean functionals of a $G$-equivariant normal test configurations of $X$ via combinatory data. Also we define a modified Futaki invariant with respect to the weight $g$, and give an expression in terms of intersection numbers. Finally we show the equivalence of different notations of stability and gives a stability criterion on $\mathbb Q$-Fano spherical varieties, which is also a criterion of existence of K\"ahler-Ricci $g$-solitons.

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	Let $G$ be a connected, complex reductive Lie group and $X$ a $\mathbb Q$-Fano $G$-spherical variety.	background 1.00	0.06	0.02	asserted 0.76
s1	In this paper we compute the weighed non-Archimedean functionals of a $G$-equivariant normal test configurations of $X$ via combinatory data.	contribution 0.79	0.95	0.10	asserted 1.00
s2	Also we define a modified Futaki invariant with respect to the weight $g$, and give an expression in terms of intersection numbers.	contribution 0.86	0.94	0.08	asserted 1.00
s3	Finally we show the equivalence of different notations of stability and gives a stability criterion on $\mathbb Q$-Fano spherical varieties, which is also a criterion of existence of K\"ahler-Ricci $g$-solitons.	result 0.90	0.95	0.82	asserted 1.00

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Original abstract

Let $G$ be a connected, complex reductive Lie group and $X$ a $\mathbb Q$-Fano $G$-spherical variety. In this paper we compute the weighed non-Archimedean functionals of a $G$-equivariant normal test configurations of $X$ via combinatory data. Also we define a modified Futaki invariant with respect to the weight $g$, and give an expression in terms of intersection numbers. Finally we show the equivalence of different notations of stability and gives a stability criterion on $\mathbb Q$-Fano spherical varieties, which is also a criterion of existence of K\"ahler-Ricci $g$-solitons.

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4 arXiv papers cite this in 5 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

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Our approach is more geometric in nature and independent of the works in [ref] [ref] .

K-stable valuations and Calabi-Yau metrics on affine spherical varieties · K-stable degeneration and K-stability of spherical cones · 0.95

Cites it as background 2 papers

In recent years, the existence problem of (singular) KE metrics and (singular) KR solitons on such $ℚ$-Fano varieties has been extensively studied (cf. [ref] , etc.).

K\"ahler-Ricci flow on $\mathbf G$-spherical Fano manifolds · Introduction · 0.96
There are several results on the existence of Kähler-Ricci solitons on Fano varieties with large symmetry, for example, on toric varieties [ref] , on spherical varieties [ref] and on $$-varieties of complexity one [ref] .

K\"ahler-Ricci solitons on Fano threefolds with non-trivial moduli · Introduction · 0.81

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Uses the method or tool of 1 paper

Then as in [ref] one can construct a non-product $G$-equivariant normal test configuration $(𝒳,ℒ)$ of $(X,-K_X)$ so that [equation] A contradiction to (4).

Finiteness of $\mathbb Q$-Fano compactifications of semisimple group with K\"ahler-Einstein metrics · The stability criterion · 0.93

Cites as support 1 paper

By [ref] , one concludes that [ref] holds if and only if $(𝒳,ℒ)$ is special after a possible base change.

Equivariant $\mathbb R$-test configurations of polarized spherical varieties · The $g$-weighted non-Archimedean functionals of $G$-equivariant normal test configurations · 0.97

Cites as background 3 papers

This was first formulated in [ref] .

On the Yau-Tian-Donaldson conjecture for generalized K\"ahler-Ricci soliton equations · Definition under the canonical lifting · 0.86
The $G$-equivariant (ample) normal test configurations of $(X,L)$ have been classified by [ref] .

2009.06463 · $G$-equivariant normal test configurations · 0.80
The $G$-equivariant (ample) normal test configurations of $(X,L)$ have been classified by [ref] .

Equivariant $\mathbb R$-test configurations of polarized spherical varieties · $G$-equivariant normal test configurations · 0.80

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