Generalized blowup of corners and fiber products
Authors:
Chris Kottke and Richard B. Melrose
Journal:
Trans. Amer. Math. Soc. 367 (2015), 651705
MSC (2010):
Primary 57R99; Secondary 14B05
Published electronically:
June 18, 2014
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Abstract: Real blowup, 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 blowup in which a new manifold and blowdown 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 blownup space. As an application of this theory, the existence of fiber products is examined for the natural smooth maps in this context, the bmaps. Transversality of the bdifferentials is shown to ensure that the settheoretic 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 bmaps 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 blowup and then has a weaker form of the universal mapping property requiring blowup of the domain.
 [DCP83]
C. De Concini and C. Procesi, Complete symmetric varieties, II; Intersection theory, Algebraic groups and related topics, Adv. Stud. Pure Math, vol. 6, Kyoto/Nagoya, 1983, pp. 481513.
 [Ful93]
William
Fulton, Introduction to toric varieties, Annals of Mathematics
Studies, vol. 131, Princeton University Press, Princeton, NJ, 1993.
The William H. Roever Lectures in Geometry. MR 1234037
(94g:14028)
 [Joy09]
Dominic
Joyce, On manifolds with corners, Advances in geometric
analysis, Adv. Lect. Math. (ALM), vol. 21, Int. Press, Somerville,
MA, 2012, pp. 225–258. MR
3077259
 [Kat94]
Kazuya
Kato, Toric singularities, Amer. J. Math. 116
(1994), no. 5, 1073–1099. MR 1296725
(95g:14056), http://dx.doi.org/10.2307/2374941
 [KKMSD73]
G.
Kempf, Finn
Faye Knudsen, D.
Mumford, and B.
SaintDonat, Toroidal embeddings. I, Lecture Notes in
Mathematics, Vol. 339, SpringerVerlag, BerlinNew York, 1973. MR 0335518
(49 #299)
 [Mel]
R. B. Melrose, Differential analysis on manifolds with corners, In preparation, partially available at http://math.mit.edu/~rbm/book.html.
 [Mel92]
Richard
B. Melrose, Calculus of conormal distributions on manifolds with
corners, Internat. Math. Res. Notices 3 (1992),
51–61. MR
1154213 (93i:58148), http://dx.doi.org/10.1155/S1073792892000060
 [Ogu06]
A. Ogus, Lectures on logarithmic algebraic geometry, Notes available at http://math.berkeley.edu/~ogus/preprints/log_book/logbook.pdf (2006).
 [Tei]
Bernard
Teissier, Monomial ideals, binomial ideals, polynomial ideals,
Trends in commutative algebra, Math. Sci. Res. Inst. Publ., vol. 51,
Cambridge Univ. Press, Cambridge, 2004, pp. 211–246. MR 2132653
(2006c:13032), http://dx.doi.org/10.1017/CBO9780511756382.008
 [DCP83]
 C. De Concini and C. Procesi, Complete symmetric varieties, II; Intersection theory, Algebraic groups and related topics, Adv. Stud. Pure Math, vol. 6, Kyoto/Nagoya, 1983, pp. 481513.
 [Ful93]
 William Fulton, Introduction to toric varieties, Annals of Mathematics Studies, vol. 131, Princeton University Press, Princeton, NJ, 1993. The William H. Roever Lectures in Geometry. MR 1234037 (94g:14028)
 [Joy09]
 D. Joyce, On manifolds with corners, Adv. Lect. Math. (ALM), 21, Int. Press, Somerville, MA, 2012. MR 3077259
 [Kat94]
 Kazuya Kato, Toric singularities, Amer. J. Math. 116 (1994), no. 5, 10731099. MR 1296725 (95g:14056), http://dx.doi.org/10.2307/2374941
 [KKMSD73]
 G. Kempf, Finn Faye Knudsen, D. Mumford, and B. SaintDonat, Toroidal embeddings. I, Lecture Notes in Mathematics, Vol. 339, SpringerVerlag, Berlin, 1973. MR 0335518 (49 #299)
 [Mel]
 R. B. Melrose, Differential analysis on manifolds with corners, In preparation, partially available at http://math.mit.edu/~rbm/book.html.
 [Mel92]
 Richard B. Melrose, Calculus of conormal distributions on manifolds with corners, Internat. Math. Res. Notices 3 (1992), 5161. MR 1154213 (93i:58148), http://dx.doi.org/10.1155/S1073792892000060
 [Ogu06]
 A. Ogus, Lectures on logarithmic algebraic geometry, Notes available at http://math.berkeley.edu/~ogus/preprints/log_book/logbook.pdf (2006).
 [Tei]
 Bernard Teissier, Monomial ideals, binomial ideals, polynomial ideals, Trends in commutative algebra, Math. Sci. Res. Inst. Publ., vol. 51, Cambridge Univ. Press, Cambridge, 2004, pp. 211246. MR 2132653 (2006c:13032), http://dx.doi.org/10.1017/CBO9780511756382.008
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Additional Information
Chris Kottke
Affiliation:
Department of Mathematics, Brown University, Providence, Rhode Island 02912
Address at time of publication:
Department of Mathematics, Northeastern University, Boston, Massachusetts 02115
Email:
ckottke@math.brown.edu, c.kottke@neu.edu
Richard B. Melrose
Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
Email:
rbm@math.mit.edu
DOI:
http://dx.doi.org/10.1090/S000299472014062223
PII:
S 00029947(2014)062223
Keywords:
Manifold with corners,
blowup,
generalized blowup,
fiber product,
monoidal complex
Received by editor(s):
April 10, 2012
Received by editor(s) in revised form:
May 15, 2013
Published electronically:
June 18, 2014
Additional Notes:
The second author was supported in part by NSF grant DMS1005944.
Article copyright:
© Copyright 2014
Copyright retained by the authors
