Higher Codimensional Alpha Invariants and Characterization of Projective Spaces

TL;DR

This paper extends alpha invariants to higher codimensions, providing bounds for K-semistable Q-Fano varieties and characterizing projective spaces through these invariants, linking them to Fano volumes.

Contribution

It introduces higher codimensional alpha invariants and uses them to characterize projective spaces among K-semistable Fano manifolds.

Findings

Lower bounds for higher codimensional alpha invariants in K-semistable Q-Fano varieties

Characterization of projective spaces using higher codimensional alpha invariants

Relation established between alpha invariants and volumes of Fano manifolds

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.