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INVARIANTS OF A COMPLEX COTANGENT LINE FIELD

We study a complex manifold M together with a smooth complex line sub-bundle E of its (1,0)-cotangent bundle. E is assumed to satisfy a certain integrability condition and a non-degeneracy condition. We attach to the structure (M,E) an invariant generalized connection on a principal bundle P over M of adapted coframes. The total space of E minus its zero section has a natural almost complex structure. We determine when it is actually a complex structure.