Delta-invariants for Fano varieties with large automorphism groups

Aleksei, Golota,

doi:10.1142/s0129167x20500779

Cited by 15 publications

(7 citation statements)

References 53 publications

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“…This follows directly from its construction. Indeed, taking a G-equivariant resolution of indeterminacy Y X × P 1 X , one realises the proper transform X0 ⊂ Y 0 as a G-invariant divisor of Y, implying that v X 0 is a G-invariant divisorial valuation on X , exactly as in Golota's proof in the Fano case [34,Proposition 3.13]. The second is that the integral test configuration associated with a G-invariant dreamy prime divisor is a G-equivariant test configuration, which, as noted by Zhu in the Fano setting [55, Theorem 3.5], follows immediately from its definition as…”

Section: Theorem 320 a Polarised Variety (X L) Is G-equivariantly Integrally K-polystable If And Only If β(F) ≥ 0 For All G-invariant Drmentioning

confidence: 90%

“…We say that a dreamy prime divisor F ⊂ Y is G-invariant if there is a G action on Y , making the map Y → X a G-invariant map, such that F is itself a G-invariant divisor on Y (by which we mean G sends F to itself rather than F being contained in the fixed point locus of G). The following is a variant of work of Golota and Zhu [34,55].…”

Section: Equivariant K-polystabilitymentioning

confidence: 99%

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Valuative stability of polarised varieties

Dervan¹,

Legendre

2022

Math. Ann.

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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 K-stability with respect to test configurations with integral central fibre. The numerical invariant governing valuative stability is modelled on Fujita’s $$\beta $$ β -invariant, 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 $$\delta $$ δ -invariant plays in the study of valuative stability and K-stability of polarised varieties.

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Section: Theorem 320 a Polarised Variety (X L) Is G-equivariantly Integrally K-polystable If And Only If β(F) ≥ 0 For All G-invariant Drmentioning

confidence: 90%

Section: Equivariant K-polystabilitymentioning

confidence: 99%

Valuative stability of polarised varieties

Dervan¹,

Legendre

2022

Math. Ann.

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show abstract

“…This follows directly from its construction. Indeed, taking a G-equivariant resolution of indeterminacy Y X × P 1 X , one realises the proper transform X0 ⊂ Y 0 as a G-invariant divisor of Y, implying that v X0 is a G-invariant divisorial valuation on X, exactly as in Golota's proof in the Fano case [31,Proposition 3.13]. The second is that the integral test configuration associated with a G-invariant dreamy prime divisor is a G-equivariant test configuration, which, as noted by Zhu in the Fano setting [50,Theorem 3.5], follows immediately from its definition as…”

Section: Proof Integrating By Parts Givesmentioning

confidence: 90%

“…divisor on Y (by which we mean G sends F to itself rather than F being contained in the fixed point locus of G). The following is a variant of work of Golota and Zhu [31,50]. Theorem 3.20.…”

Section: Proof Integrating By Parts Givesmentioning

confidence: 97%

Valuative stability of polarised varieties

Dervan¹,

Legendre²

2020

Preprint

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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.

show abstract

“…But we will take a purely algebraic approach, again making use of the C * -action. Indeed, by [BJ17,Gol19], to compute δ, it is enough to consider C * -invariant divisorial valuations, which can greatly simplify the computation. 1.2.…”

mentioning

confidence: 99%