Modification of a complex space

Summary

A modification of a reduced complex space $Y$ is a proper surjective holomorphic map $X \to Y$ that is biholomorphic outside thin analytic subsets. It changes $Y$ only over a nowhere dense analytic set, where it may replace points by larger sets. Blowing up, normalization and resolution of singularities give modifications. Modifications are the analytic counterpart of proper birational morphisms in algebraic geometry.