Generalized blow-up of corners and fiber products
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Melrose_Generalized blow-up.pdf
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620.39 KB
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Author(s)
Kottke, Chris
Melrose, Richard B
Date Issued
June 2014
Journal
Transactions of the American Mathematical Society
Citation
Kottke, Chris, and Richard B. Melrose. “Generalized Blow-up of Corners and Fiber Products.” Transactions of the American Mathematical Society 367.1 (2014): 651–705.
Version
Final published version
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.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use.
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DOI of Published Version
https://doi.org/10.1090/S0002-9947-2014-06222-3
