Projective Line P1 Research Articles

A characterization of the Veronese varieties

Let Pm(C) be the complex projective space of dimension m. In a previous paper [2] we have provedTHEOREM A. Let f be a Kaehlerian immersion of a connected, complete Kaehler manifold Mn of dimension n into Pm(C). If the image f(τ) of each geodesic τ in Mn lies in a complex projective line P1(C) of Pm(C), then f(Mn) is a complex projective subspace of Pm(C), and f is totally geodesic.

A class of indecomposable algebraic vector bundles

In 1956, Grothendieck [4] proved, that every n-dimensional holomorphic vector bundle on the projective line P1 over an algebraically closed field is the direct sum of n line bundles. In the same paper he raised the question whether P1 is the only projective variety with this property. In a talk in Oberwolfach in 1962, van de Ven proved this for non-singular varieties by showing that the restriction of the tangent bundle of the projective space P, to a non-singular algebraic sub-variety is not the direct sum of n line bundles, unless the variety is a rational curve. In this paper the following generalization of this theorem is given: