Generalized blow-up of corners and fiber products

Abstract

Real blow-up, including inhomogeneous versions, of boundary faces of a manifold (with corners) is an important tool for resolving singularities, degeneracies and competing notions of homogeneity. These constructions are shown to be particular cases of generalized boundary blow-up in which a new manifold and blow-down map are constructed from, and conversely determine, combinatorial data at the boundary faces in the form of a refinement of the basic monoidal complex of the manifold. This data specifies which notion of homogeneity is realized at each of the boundary hypersurfaces in the blown-up space. As an application of this theory, the existence of fiber products is examined for the natural smooth maps in this context, the b-maps. Transversality of the b-differentials is shown to ensure that the set-theoretic fiber product of two maps is a binomial variety. Properties of these (extrinsically defined) spaces, which generalize manifolds but have mild singularities at the boundary, are investigated, and a condition on the basic monoidal complex is found under which the variety has a smooth structure. Applied to b-maps this additional condition with transversality leads to a universal fiber product in the context of manifolds with corners. Under the transversality condition alone the fiber product is resolvable to a smooth manifold by generalized blow-up and then has a weaker form of the universal mapping property requiring blow-up of the domain.