Vector Fields on Smooth Threefolds Vanishing on Complete Intersections

Details Vector Fields on Smooth Threefolds Vanishing on Complete Intersections Vector Fields on Smooth Threefolds Vanishing on Complete Intersections

2001-04-24

arxiv.org/abs/math/0104217v1

Mathematics

Algebraic Geometry

11 pages

Scientific paper

The existence of a vector field on a compact Kaehler manifold with nonempty zero locus and the properties of this zero locus strongly influence the geometry of the manifold. For example, J. Wahl proved that the existence of a vector field vanishing on an ample divisor of a projective normal variety X implies that X is a cone over this divisor. If X is smooth, X will be isomorphic to the n-dimensional projective space. This paper is a first attempt to generalize Wahl's theorem to higher codimensions: Given a complex smooth projective threefold X and a vector field on X vanishing on an irreducible and reduced curve which is the scheme theoretic intersection of two ample divisors, X is isomorphic to the 3-dimensional projective space or the 3-dimensional quadric.

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