It's a classic case of most papers using a concept and proving high-level mathematics about it, without ever stating the definition or some simple properties of the object. Which is just peachy for the complex geometry novice learner.
Would anyone here happen to have this sort of mapping in their daily toolbox and be comfortable sharing their knowledge with me?
Let M, N be compact, complex manifolds. We call f: M\rightarrow N a proper modification if f is a holomorphic mapping which is proper, and if N contains an analytic set Y whose preimage f^{-1}(Y)=:E (called the "exceptional set") is a hypersurface, and such that the restriction of f to M-E (mapping onto N-Y) is a biholomorphism.