Classification of weighted dual graphs with only complete intersection singularities structures
Authors:
Fan Chung, Yi-Jing Xu and Stephen S.-T. Yau
Journal:
Trans. Amer. Math. Soc. 361 (2009), 3535-3596
MSC (2000):
Primary 32S25, 58K65, 14B05
DOI:
https://doi.org/10.1090/S0002-9947-09-04524-3
Published electronically:
March 4, 2009
MathSciNet review:
2491891
Full-text PDF
Abstract | References | Similar Articles | Additional Information
Abstract: Let be normal singularity of the 2-dimensional Stein space
. Let
be a minimal good resolution of
, such that the irreducible components
of
are nonsingular and have only normal crossings. Associated to
is weighted dual graph
which, along with the genera of the
, fully describes the topology and differentiable structure of
and the topological and differentiable nature of the embedding of
in
. In this paper we give the complete classification of weighted dual graphs which have only complete intersection singularities but no hypersurface singularities associated to them. We also give the complete classification of weighted dual graphs which have only complete intersection singularities associated with them.
- [Ar] M. Artin, On isolated rational singularities of surfaces, Amer. J. Math. 88 (1996), 129-136. MR 0199191 (33:7340)
- [Ba] H. Bass, On the ubiquity of Gorenstein rings, Math. Z. 82 (1963), 8-28. MR 0153708 (27:3669)
- [Gr] H. Grauert, Über Modifikationen und exzeptionelle analytische Mengen, Math. Ann. 146 (1962), 331-368. MR 0137127 (25:583)
- [Gr-Ri] H. Grauert and O. Riemenschneider, Verschwindungssätze für analytische Kohomologiegruppen auf komplexen Räumen, Invent. Math. 11 (1970), 263-292. MR 0302938 (46:2081)
- [HNK] F. Hirzebruch, W. Neumann, and S. Koh, Differentiable manifolds and quadratic forms, Lecture Notes in Pure and Applied Mathematics, vol. 4, Marcel Dekker, New York, 1971. MR 0341499 (49:6250)
- [La1] H. Laufer, Normal two-dimensional singularities, Annals of Math. Studies, no. 71, Princeton University Press, Princeton, NJ, 1971. MR 0320365 (47:8904)
- [La2] -, On rational singularities, Amer. J. Math. 94 (1972), 597-608. MR 0330500 (48:8837)
- [La3] -, Deformations of resolutions of two-dimensional singularities, Rice University Studies 1 (1973), no. 59, 53-96. MR 0367277 (51:3519)
- [La4] -, On minimally elliptic singularities, Amer. J. Math. 99 (1977), 1257-1295. MR 0568898 (58:27961)
- [Mi] J. Milnor, Singular points of complex hypersurfaces, Ann. Math. Studies, no. 61, Princeton University Press, Princeton, NJ, 1968. MR 0239612 (39:969)
- [Ne] W. Neumann, A calculus for plumbing applied to the topology of complex surface singularities and degenerating complex curves, Trans. A.M.S. 268 (1981), 299-344. MR 632532 (84a:32015)
- [Se] J.-P. Serre, Groupes algébriques et corps de classes, Actualités Scientifiques et Industrielles, no. 1264, Hermann, Paris, 1959. MR 0103191 (21:1973)
- [Wa] P. Wagreich, Elliptic singularities of surfaces, Amer. J. Math. 92 (1970), 419-454. MR 0291170 (45:264)
- [Ya1] S. S.-T. Yau, Normal singularities of surfaces, Proc. Sympos. Pure Math. 72 (1978), 195-198. MR 520537 (80f:14020)
- [Ya2] -, On maximally elliptic singularities, Trans. A.M.S. 257 (1980), 269-329. MR 552260 (80j:32021)
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Additional Information
Fan Chung
Affiliation:
Department of Mathematics, University of California at San Diego, La Jolla, California 92093-0112
Yi-Jing Xu
Affiliation:
Department of Mathematics, John Tyler Community College, 13101 Jefferson Davis Highway, Chester, Virginia 23831-5316
Stephen S.-T. Yau
Affiliation:
Department of Mathematics, Statistics, and Computer Science (M/C 249), University of Illinois at Chicago, 851 South Morgan Street, Chicago, Illinois 60607-7045
Email:
yau@uic.edu
DOI:
https://doi.org/10.1090/S0002-9947-09-04524-3
Received by editor(s):
March 2, 2007
Published electronically:
March 4, 2009
Additional Notes:
The third author’s research was partially supported by an NSF grant.
Dedicated:
Dedicated to Henry Laufer on the occasion of his 65th birthday
Article copyright:
© Copyright 2009
American Mathematical Society