Birational superrigidity is not a locally closed property

Zhuang, Ziquan

doi:10.1007/s00029-020-0536-1

Cited by 3 publications

(2 citation statements)

References 37 publications

(29 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Extending the birigidity, tririgidity and so on notion of solid Fano variety is introduced in [AO18]: A Fano variety of Picard number 1 is solid if any Mori fiber space in the birational equivalence class is a Fano variety of Picard number 1. Solid Fano varieties are expected to behave nicely in moduli ( [Zhu20a]). Only some evidence is known ([KOW19]) for the following question.…”

Section: 4a Generalizations Of the Conjecturementioning

confidence: 99%

K-stability of birationally superrigid Fano 3-fold weighted hypersurfaces

Kim,

Okada,

Won

2023

Forum of Mathematics, Sigma

View full text Add to dashboard Cite

We prove that the alpha invariant of a quasi-smooth Fano 3-fold weighted hypersurface of index $1$ is greater than or equal to $1/2$ . Combining this with the result of Stibitz and Zhuang [SZ19] on a relation between birational superrigidity and K-stability, we prove the K-stability of a birationally superrigid quasi-smooth Fano 3-fold weighted hypersurfaces of index $1$ .

show abstract

Section: 4a Generalizations Of the Conjecturementioning

confidence: 99%

K-stability of birationally superrigid Fano 3-fold weighted hypersurfaces

Kim,

Okada,

Won

2023

Forum of Mathematics, Sigma

View full text Add to dashboard Cite

show abstract

“…Extending bi-rigidity, tri-rigidity and so on, notion of solid Fano variety in introduced in [AO18]: a Fano variety of Picard number 1 is solid if any Mori fiber space in the birational equivalence class is a Fano variety of Picard number 1. Solid Fano varieties are expected to behave nicely in moduli ( [Zhu20a]). Only a few evidences are known ([KOW19]) for the following question.…”

Section: Existence Of Kähler-einstein Metricsmentioning

confidence: 99%

K-stability of birationally superrigid Fano 3-fold weighted hypersurfaces

Kim,

Okada,

Won

2020

Preprint

View full text Add to dashboard Cite

We prove that the alpha invariant of a quasi-smooth Fano 3-fold weighted hypersurface of index 1 is greater than or equal to 1/2. Combining this with the result of Stibitz and Zhuang [SZ19] on a relation between birational superrigidity and K-stability, we prove the K-stability of a birationally superrigid quasi-smooth Fano 3-fold weighted hypersurfaces of index 1. ContentsChapter 1. Introduction 1.1. K-stability, birational superrigidity, and a conjecture 1.2. Evidences for the conjecture 1.3. Main results Chapter 2. Preliminaries 2.1. Basic definitions and properties 2.2. Weighted projective varieties 2.3. The 95 families Chapter 3. Methods of computing log canonical thresholds 3.1. Auxiliary results 3.2. Methods Chapter 4. Smooth points 4.1. Smooth points on U 1 for families indexed by I \ I 1 4.2. Smooth points on H x \ L xy for families with 1 < a 1 < a 2 4.3. Smooth points on L xy for families with a 1 < a 2 , Part 1 4.4. Smooth points on L xy for families with a 1 < a 2 , Part 2 4.5. Smooth points on H x for families with 1 < a 1 = a 2 Chapter 5. Singular points 5.1. Non-BI centers 5.2. EI centers 5.3. Equations for QI centers 5.4. QI centers: exceptional case 5.5. QI centers: degenerate case 5.6. QI centers: non-degenerate case Chapter 6. Families F 2 , F 4 , F 5 , F 6 , F 8 , F 10 and F 14 6.1. Families F 6 , F 10 and F 14 6.2. The family F 2 6.3. The family F 4 6.4. The family F 5 6.5. The family F 8 Chapter 7. Further results and discussion on related problems 7.1. Birationally superrigid Fano 3-folds of higher codimensions 7.2. Lower bound of alpha invariants 7.3. Existence of Kähler-Einstein metrics 7.4. Birational rigidity and K-stability 7.5. Further computations of alpha invariants Chapter 8. The table Bibliography CHAPTER 1Recently Stibitz and Zhuang relaxed the assumption on the alpha invariants significantly under the assumption of birationally superrigidity, and obtained the following.Theorem 1.2.5 ([SZ19, Theorem 1.2, Corollary 3.1]). Let X be a birationally superrigid Fano variety. If α(X) ≥ 1/2, then X is K-stable.Note that the assumption on the alpha invariant is α(X) > 1/2 in [SZ19, Theorem 1.2], but the equality is allowed by [SZ19, Corollary 3.1]. It is informed by C. Xu and Z. Zhuang that one can even conclude the uniform K-stability of X in Theorem 1.2.5 under the same assumption. The notion of uniform K-stability is originally introduced in [Der16b] and [BHJ17] (see also [Fuj19b] and [BJ20]) and it is stronger than K-stability. Moreover it is very important to mention that uniform K-stability implies the existence of a KE metric ([LTW19]). Combining these results, we have the following.Theorem 1.2.6 ([Xu20, Theorem 9.6], [SZ19], [LTW19]). Let X be a birationally superrigid Fano variety and assume that α(X) ≥ 1/2. Then X is uniformly K-stable. In particular, X is K-stable and it admits a KE metric. Main resultsWe state Main Theorem of this article.Theorem 1.3.1 (Main Theorem). Let X be a quasi-smooth Fano 3-fold weighted hypersurface of index 1. Then α(X) ≥ 1/2.The following is a dire...

show abstract

Seshadri constants and K-stability of Fano manifolds

Abban

Zhuang

2023

Duke Math. J.

View full text Add to dashboard Cite

Birational superrigidity is not a locally closed property

Cited by 3 publications

References 37 publications

K-stability of birationally superrigid Fano 3-fold weighted hypersurfaces

K-stability of birationally superrigid Fano 3-fold weighted hypersurfaces

K-stability of birationally superrigid Fano 3-fold weighted hypersurfaces

Seshadri constants and K-stability of Fano manifolds

Contact Info

Product

Resources

About