Über die Grassmannmannigfaltigkeit \(G(1,4)\) und verwandte Mannigfaltigkeiten
A compact complex manifold V of dimension n with an ample line bundle L, is called (following Fujita) a Del Pezzo manifold of degree 5 if \(c_ 1(L)^ n=5\) and \(K_ V+(n-1)L=0\), where \(K_ V\) is the canonical divisor. Examples of such manifolds for \(n=1,2,3,4,5,6\) are obtained as intersections in \({\mathbb{P}}^ 9\) of a general linear subspace of codimension 6-n with the Grassmann manifold G(1,4) of lines in \({\mathbb{P}}^ 4\) considered as a submanifold of \({\mathbb{P}}^ 9\) via the Plücker embedding. Fujita has shown that: Up to biholomorphic equivalence there are no other Del Pezzo manifolds of degree 5. The crucial case \(n=3\) of Fujita's result is due to Iskovskikh. In the present article the author gives a different and substantially simplified proof of Fujita's result and also obtains (what he calls) Moishezon variants of the result.
scientific article; zbMATH DE number 3927096