Products of manifolds with fibered corners

Abstract

Manifolds with fibered corners arise as resolutions of stratified spaces, in many body compactifications of vector spaces, moduli spaces, and other settings. We define a category of fibered corners manifolds which has products and transverse fiber products, generalizing both the resolutions of products of stratified spaces and many body products, which are special cases. The product in the fibered corners category is a resolution of the cartesian product by blow-up which we call the 'ordered product'. This ordered product is a natural product for wedge (aka incomplete edge) metrics and quasi-fibered boundary metrics, a class which includes QAC and QALE metrics. Contents 1. Introduction 2. Background 2.1. Normal Models 2.2. Generalized blow-up 2.3. Rational combinations of boundary defining functions 3. Ordered corners 3.1. Products 3.2. Boundary hypersurfaces 3.3. Fiber products 4. Fibered corners 4.1. Products and fiber products 5. Relation to other theories 5.1. Smoothly stratified spaces 5.2. Many body spaces 6. Geometric structures 6.1. Products of Φ structures 6.2. Products of wedge structures 7. Equivalence of minimal and maximal joins Appendix A. Tube systems and fibrations References