A Remark on Characterizations of Projective Spaces among Singular Varieties

Abstract

Let X be a projective variety of dimension n ≥ 2 with at worst log-terminal singularities and let \(E \subseteq T_X\) be an ample vector bundle of rank r. By partially extending previous results due to Andreatta and Wiśniewski in the smooth case, we prove that if r = n then \(X \cong {\mathbb{P}}^n\), while if r = n − 1 and X has only isolated singularities, then either \(X \cong {\mathbb{P}}^n\) or n = 2 and X is the quadric cone Q 2.