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Vector bundles on ample divisors

1981; Mathematical Society of Japan; Volume: 33; Issue: 3 Linguagem: Inglês

10.2969/jmsj/03330405

1881-1167

Takao Fujita,

Homotopy and Cohomology in Algebraic Topology

Suppose that a scheme $A$ lies as an ample divisor in another scheme $M$ .Then, as we saw in [5] and [2], the structure of $M$ is closely related to that of $A$ .Keeping this principle in mind, we study in \S 1 the behaviour of a vector bundle $F$ on $M$ in relation to that of $F_{A}$ .In \S 2 and \S 3 we prove the following extendability criterion announced in [1]: A vector bundle $E$ can be extended to a vector bundle on $MifH^{2}(A, e_{nd}(E)[-tA])=0foranyt\geqq 1,$ $H^{p}(A,$ $ E[tA]\rangle$ $=0$ for any $0<P<\dim A,$ $t\in Z$ and if $M$ is non-singular.In \S 4 and \S 5, as an application, we show that the Grassmann variety $G_{n.r}$parametrizing r-dimen- sional linear subspaces of an n-dimensional vector space cannot be an ample divisor in any manifold except the well known classical cases, namely the cases in which $r=1,$ $r=n-1$ or $(n, r)=(4,2)$ .Notation, Convention and Terminology.In this paper we fix once for all an algebraically closed field $k$ of any characteristic and assume that everything is defined over $k$ .Basically we employ the same notation as in [2].In particular, vector bundles are confused with the locally free sheaves of their sections.Here we show examples of symbols.