Higher codimensional alpha invariants and characterization of projective spaces

Abstract

We generalize the definition of alpha invariant to arbitrary codimension. We also give a lower bound of these alpha invariants for K-semistable Q-Fano varieties and show that we can characterize projective spaces among all K-semistable Fano manifolds in terms of higher codimensional alpha invariants. Our results demonstrate the relation between alpha invariants of any codimension and volumes of Fano manifolds in the characterization of projective spaces.

Noether-Fano Inequalities and Canonical Thresholds on Fano Varieties

“…For the global log canonical threshold this is known to be true by [1]. On the other hand, for global movable log canonical thresholds this is still unknown [13]. In the case of canonical thresholds it does not seem to follow immediately from Theorem 1.5 and boundedness of terminal Fano varieties.…”

“…The global log canonical threshold is equivalent to Tian's alpha invariant used to show the existence of Kähler-Einstein metrics on Fano varieties [12]. The movable version and its relation to K-stability has been studied in [11] and [13].…”

Birational superrigidity is not a locally closed property

“…This paper is organized as follows. Birational superrigidity can be characterized by the singularities of movable boundaries and its openness is closely related to the semicontinuity of higher codimensional alpha invariants recently introduced by [Zhu18a], thus in §2 we study some properties of these invariants which may be of independent interest. Theorems 1.2 and 1.3 are proved in §3 and §4 respectively.…”