On the complex Grassmann manifold (Q1109845)

This paper proves a characterization of the Grassmann manifold \(G=G(n,m+n)\) which generalizes in a stronger form the former result of \textit{F. Hirzebruch} and \textit{K. Kodaira} [J. Math. Pures Appl., IX Sér. 36, 201-216 (1957; Zbl 0090.386)] about \({\mathbb{P}}^ n({\mathbb{C}})\), recognized by its cohomology ring and the existence of a positive line bundle L such that: \(c_ 1(L)\) generates \(H^ 2({\mathbb{Z}})\), and \(h^ 0(L^{\nu})=\left( \begin{matrix} n+\nu \\ n\end{matrix} \right).\) The proof by Hirzebruch cannot be directly adapted; one needs the following lemma: Let \(\pi:\quad G\to {\mathbb{P}}^ N({\mathbb{C}})\) be the Plücker embedding; let \(V\varsubsetneq G\) be an irreducible subvariety not contained in any hyperplane. Then: \(\deg (V)>\deg (G)\).