Cohomological characterizations of projective spaces and hyperquadrics (Q958171)

Projective spaces and hyperquadrics are among the simplest projective manifolds and characterizing them in terms of properties of their tangent bundles is an important subject in algebraic geometry. The most famous example is certainly \textit{S. Mori}'s proof of the Hartshorne conjecture saying that the projective space is the only projective manifold with ample tangent bundle [Ann. Math. (2) 110, 593--606 (1979; Zbl 0423.14006)]. Similarly \textit{K. Cho} and \textit{E. Sato} characterize hyperquadrics and projective spaces as the projective manifolds \(X\) such that \(\bigwedge^2 T_X\) is ample [J. Math. Kyoto Univ. 35, No. 1, 1--33 (1995; Zbl 0832.14031)]. More recently the focus has been on obtaining characterizations assuming only the existence of subsheaves with certain positivity properties: \textit{M. Andreatta} and \textit{J. Wiśniewski} have shown that if \(X\) is a complex projective manifold of dimension \(n\) such that \(T_X\) contains an ample locally free subsheaf \(E\), then \(X\) is the projective space and \(E\) the tangent bundle or a sum of hyperplane divisors [Invent. Math. 146, No. 1, 209--217 (2001; Zbl 1081.14060)]. \newline In the paper under review the authors show a statement that can be seen as an interpolation between the theorems of Cho-Sato and Andreatta-Wiśniewski: let \(X\) be a complex projective manifold of dimension \(n\), and let \(L\) be an ample line bundle on \(X\). If for some positive integer \(p\), the vector bundle \(\bigwedge^p T_X \otimes L^{-p}\) has a non-zero global section, then \(X\) is a projective space or a hyperquadric and \(L\) is the hyperplane divisor. This generalizes a theorem of \textit{J.M. Wahl} [Invent. Math. 72, 315--322 (1983; Zbl 0544.14013)] dealing with the case \(p=1\). The proof proceeds by studying the minimal rational curves on \(X\) and their associated rationally connected quotients.