Modifications

Abstract

This chapter is devoted to the study of the bimeromorphic geometry of complex spaces. Bimeromorphic geometry is the study of complex spaces up to bimeromorphic equivalence. Roughly speaking, two complex spaces X,Y are bimeromorphically equivalent if they are isomorphic outside thin analytic sets. If X and Y are irreducible, this means that their fields of meromorphic functions are isomorphic. For two given spaces X and Y which are bimeromorphically equivalent, one can find another complex space Z and a diagram with γ the bimeromorphic equivalence described above and α, β holomorphic everywhere defined maps which are isomorphisms almost everywhere. The maps α and β are called modifications. So the study of bimeromorphic geometry is the study of modifications of complex spaces. To modify a complex space means to take out an analytic set and to substitute it by some other analytic set; so this procedure is a kind of surgery. In chap. V we have already met several modifications: the Remmert reductions of 1-convex spaces. In our terminology 1-convex spaces are just the modifications of Stein spaces in a discrete set D: D is taken out of the Stein space X and a higher dimensional set A (the exceptional set) is put in instead. Of course A cannot be arbitrary, as there are restrictions due to the local geometry of X. In dimension 1 there is not enough space for interesting bimeromorphic geometry. In dimension ≥ 2 things change completely; there is a rich bimeromorphic geometry. The most basic example of a modification is the blow-up of a point x ∈ X in a complex manifold of dimension n. The point x can be replaced by a projective space ℙ _{n −1} which can be viewed as the space of all tangent directions in x. In particular, this blow-up separates all curves in X meeting transversally in x.