Smooth projective varieties with the ample vector bundle \(\overset {2}\bigwedge T_ X\) in any characteristic
Let \(X\) be an \(n\)-dimensional smooth projective variety such that \(\bigwedge^2 T_X\) is ample. Here the authors prove that if either \(n \geq 5\) or \(n \geq 3\) and the algebraically closed base field has characteristic 0, then \(X\) is isomorphic either to \(\mathbb{P}^n\) or to a hyperquadric. The proof (as usual in similar type of results) uses Mori theory to produce families of rational curves contained in \(X\). However, due to positive characteristic technical problems, the authors must be very skilful; the trouble comes from cuspidal rational curves. The authors use also heavily the Fulton-Hansen connectedness theorem. They start an attempt to classify Fano manifolds in terms of their length: \(\text{length} (X) : = \min \{(-K_X \cdot C) |C\) is a rational curve in \(X\}\).
scientific article; zbMATH DE number 784731