Serre’s Construction of Rank Two Vector Bundles and the Transversal Jet Bundles of Certain Codimension One Holomorphic Foliations

Let S be a locally complete intersection subvariety of ℂℙ^d of pure dimension d − 2 . Serre gave a construction of a rank two vector bundle over ℂℙ^d under the assumption that the determinant of the normal bundle to S extends to ℂℙ^d. We give an explicit description of this rank two vector bundle in the special case where S is the singular locus of a codimension one foliation on a complex manifold M. This foliation is assumed to satisfy certain nondegeneracy conditions. The rank two vector bundle is obtained from certain transversal jet bundles, associated to a foliation, that are constructed here.

AMS (2000) Subject Classification. 13C10, 13B25.