For a divisor in a complex manifold, what is known about a complementary bundle to the divisor in the manifold (either for the tangent or the cotangent bundle). Is there a description in terms of holomorphic line bundles? For a Kahler manifold it would be possible just to take the normal bundle, and split this into two line bundles over the divisor by taking the eigenspaces of the complex structure map $J$ (hope I got this right). But then what properties do these line bundles have, and how do they relate to the properties of the divisor? As a learner in algebraic geometry, is there somewhere to look to find out about this?
Secondly an apology, I am definitely not an expert in algebraic geometry so I apologise if the answer is well known (it likely is) or if my question is confused (especially the latter)! I will have to explain where it is from: I would like to look at divisors on a noncommutative complex manifold. In noncommutative differential geometry ideas of rank or codimension are rather difficult, and in general require adding extra structure. (Understandably, textbooks on classical 'commutative' manifolds are happy simply saying complex codimension one, as there is not any problem in that case.) However the rank one case (line bundles) is known in noncommutative geometry - they have several definitions as invertible bimodules, self Morita equivalences and as integer graded Hopf Galois extensions. Even the idea of 'rank 2' is rather difficult to pin down, thus my wish to try to `define' codimension one by having a complementary bundle described in terms of line bundles of known properties.
