Ampleness in complex homogeneous spaces and a second Lefschetz theorem

Goldstein, Norman

doi:10.2140/pjm.1983.106.271

Cited by 3 publications

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“…One might also expect that if Y = G/P, then X should be almost homogeneous. In case Y is a Grassmannian or a quadric, this has been checked by Goldstein [Go83]. Of course, if Y = P n , then X is even homogeneous.…”

Section: Conclusion and Open Questionsmentioning

confidence: 99%

Contact Kähler Manifolds: Symmetries and Deformations

Peternell

Schrack

2014

Algebraic and Complex Geometry

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We study complex compact Kähler manifolds X carrying a contact structure. If X is almost homogeneous and b 2 (X) ≥ 2, then X is a projectivised tangent bundle (this was known in the projective case even without assumption on the existence of vector fields). We further show that a global projective deformation of the projectivised tangent bundle over a projective space is again of this type unless it is the projectivisation of a special unstable bundle over a projective space. Examples for these bundles are given in any dimension.

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Section: Conclusion and Open Questionsmentioning

confidence: 99%

Contact Kähler Manifolds: Symmetries and Deformations

Peternell

Schrack

2014

Algebraic and Complex Geometry

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Conull hypersurfaces in Minkowski space

Goldstein¹

1982

Proc. Amer. Math. Soc.

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Abstract. A submanifold of M = Gr(2, C4) is conull when its conormal space is in the kernel of the dualized conformai metric of M. We show that there are no conull compact complex 3-dimensional submanifolds of M.Let M be complex Minkowski space, complex analytically homeomorphic to Gr(2, C4), the Grassmannian of 2-planes in C4.Each linear P2 in M is a null manifold i.e. the tangent space TX(P2) consists entirely of null vectors, for each x E P2 cf. Let A be a complex submanifold of M. We say that A is conull if the conormal space of X,N*XC T*M, consists entirely of null covectors. It is an exercise in linear algebra to verify that for dim(A) = 2, Ais null precisely when Ais conull. However, the plethora of null curves is in marked contrast to the lack of conull hypersurfaces.Theorem. Let X be a smooth compact complex hypersurface of M. Then X is not conull.Proof. Let G = PGL(3, C) be the projective linear group. The group G acts transitively on M, and defines an action on the cotangent bundle, T*M. This latter action has 3 orbits-the zero section, the null covectors, and the open orbit. It is well known that the normal bundle, TVA, of A in M is ample (e.g. [3, 2.9]

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Ampleness and connectedness in complex $G/P$

Goldstein¹

1982

Trans. Amer. Math. Soc.

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Ampleness in complex homogeneous spaces and a second Lefschetz theorem

Cited by 3 publications

References 13 publications

Contact Kähler Manifolds: Symmetries and Deformations

Contact Kähler Manifolds: Symmetries and Deformations

Conull hypersurfaces in Minkowski space

Ampleness and connectedness in complex $G/P$

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