In the book of Olivier Debarre (Tores et variétés abéliennes complexes), at the moment where he defines divisors (p. 40), he says that one could define a divisor of a meromorphic function as a formal sum of codimension 1 sets (which we can define easily with the associated Hausdorff measure on the complex manifold) at the cost of some technical difficulties, which are solved by an abstract definition.


Therefore I wanted to know if the alternative construction (by formal sums) was actually used and if there was a good reference for it.



Definition of divisors on complex manifolds
I don't see the role of Hausdorff measure here. You can define codimension 1 sets by their being the closed sets which are locally identified in charts with codimension 1 sets, i.e. with zero loci of analytic functions. This is standard, but be careful of the distinction between Weil divisors and Cartier divisors on singular varieties. This is all in Huybrechts, [Complex Geometry](https://books.google.ie/books/about/Complex_Geometry.html?id=sWbd0rE3mhIC&redir_esc=y), p. 77.


