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There is a canonical way to construct holomorphic vector fields on $\mathbb{C}P^2,$ and that way is described in Zoladek's "Monodromy Group", page 335. If you read the description, it will be pretty clear what the index is (note that if the vector field is holomorphic, it is given by at most quadratic polynomials, otherwise there are poles. There is a large literature [much of it by Iliashenko] studying such fields and the associated foliations.