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...