K-semistable Fano manifolds with the smallest alpha invariant

Abstract

Abstract. In this short note, we show that K-semistable Fano manifolds with the smallest alpha invariant are projective spaces. Singular cases are also investigated.

On the sharpness of Tian's criterion for K-stability

“…Then the moduli stack of Q-Fano varieties X satisfying α(X) > 1 2 of dimension n and volume V is represented by a separated Deligne-Mumford stack of finite type, which has a coarse moduli space that is a separated algebraic space. Indeed, boundedness follows from [Jia17]; openness follows from [BL18b, Theorem B]; separatedness follows from Proposition 3.3. We believe such moduli spaces are interesting and deserve further investigation.…”

“…Indeed X 0 has the smallest αinvariant among all K-semistable Q-Fano varieties by [FO18,Theorem 3.5]. These Q-Fano varieties have been studied by C. Jiang [Jia17] where he showed that P n is the only Ksemistable Fano manifold with the smallest α-invariant. We provide a full characterization of such Q-Fano varieties to complement Jiang's results (see Theorem 3.9).…”

“…We provide a full characterization of such Q-Fano varieties to complement Jiang's results (see Theorem 3.9). We ask the following question about the sharpness of Tian's criterion and Jiang's conjecture [Jia17,Conjecture 1.6].…”

On the Sharpness of Tian’s Criterion for K-Stability

“…In particular, Aut(X) is finite for a Q-Fano variety X satisfying α(X) > 1 2 . We ask the following question about the sharpness of Tian's criterion and Jiang's conjecture [Jia17b,Conjecture 1.6].…”

Higher codimensional alpha invariants and characterization of projective spaces

“…When k = 1, Theorem B reduces to the main result in [Jia17] that characterizes projective spaces among all K-semistable Fano manifolds in terms of the alpha invariant. When k = n, Theorem B reduces to the following result: Fuj18]).…”