Ampleness and Connectedness in Complex G/P

Goldstein, Norman

doi:10.2307/1999515

Cited by 4 publications

(7 citation statements)

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“…Then k < t + m. For the Grassmannian, Gr (n, r), of the quotient C r 's of C n , t = r(n -r) -n + 1 and for the any smooth quadric t = 1 (see [5,7]). For the general formula see [3].…”

mentioning

confidence: 99%

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Homotopy groups of pullbacks of varieties

Sommese

Ven

1986

Nagoya Mathematical Journal

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In [2, § 9] there is a general result of Fulton and Lazarsfeld relating the homotopy groups of a subvariety of Pc in a certain range of dimensions with those of its pullback under a holomorphic map in the corresponding range of dimensions. It is asked in [2, § 10] whether here is a corresponding result with P n c replaced by a general rational homogeneous manifold, Y, and with the range of dimensions alluded to above shifted by the ampleness of the holomorphic tangent bundle of Y in the sense of [4]. In this paper we use the techniques of [4,5,6,7] to answer this question in the affirmative.Let us first recall the notion of ^-ampleness for holomorphic vector bundles [4; see 1 also]. When k -0 this notion coincides with ampleness in the sense of Grothendieck-Hartshorne. Since all the bundles for which we need this notion are spanned, the definition takes a very simple form. Let £ be a holomorphic vector bundle on a compact complex manifold that is spanned at all points by global holomorphic sections. E is k-ample if for each subvariety Z ci X such that E\ z has a trivial quotient bundle, it is true that dim Z < k.

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“…Then k < t + m. For the Grassmannian, Gr (n, r), of the quotient C r 's of C n , t = r(n -r) -n + 1 and for the any smooth quadric t = 1 (see [5,7]). For the general formula see [3].…”

mentioning

confidence: 99%

“…There is a whole literature on connectedness results (see [2]). In particular for general Y as above, Faltings [1] has a connectedness result that allows W to be singular; there is a discussion of this in [3].…”

mentioning

confidence: 99%

Homotopy groups of pullbacks of varieties

Sommese

Ven

1986

Nagoya Mathematical Journal

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“…3.1 An important class of homogeneous vector bundles is provided by the tangent bundles of homogeneous projective manifolds G/P, where G is a semisirnple complex Lie group and P is a parabolic subgroup. The ampleness of such bundles was first worked out by Goldstein [5]. We now give a summary of his results to illustrate how the ampleness formula §2 can be applied.…”

Section: Tangent Bundlesmentioning

confidence: 95%

“…In §3 we review the results of Goldstein [5]. His calculation of the ampleness of the tangent bundle of X = G / P was one of the main inspirations for the generalization to arbitrary homogeneous vector bundles in this paper.…”

Section: On the Ampleness Of Homogeneous Vector Bundles By Dennis M mentioning

confidence: 99%