2001/04/24 by Thomas Eckl, Eckl, Thomas
Mathematics · #14F05 (Secondary) #14L99 #14M20 (Primary) #Algebraic Geometry (math.AG) #FOS: Mathematics #math.AG #msc:14F05 #msc:14L99 #msc:14M20
paper · pdf · doi:10.48550/arxiv.math/0104217
11 pages
arxiv created 2001/04/24 · arxiv updated 2009/11/30
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.