On the monodromy at isolated singularities of weighted homogeneous polynomials
Author:
Benjamin G. Cooper
Journal:
Trans. Amer. Math. Soc. 269 (1982), 149166
MSC:
Primary 32C40; Secondary 14B05
MathSciNet review:
637033
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Abstract: Assume is a weighted homogeneous polynomial with isolated singularity, and define by . If the monomials of are algebraically independent, then the closure of in admits a deformation into the subset where each monomial of has nonnegative real values. For the polynomial , is a cell complex of dimension , invariant under a characteristic map of the fibration , and the inclusion induces isomorphisms in homology. To compute the homology of the link it thus suffices to calculate the action of on . Let . Let be the weights associated with , satisfying for and . Let , , . Then and .
 [1]
V.
I. Arnol′d, Normal forms of functions in the neighborhood of
degenerate critical points, Uspehi Mat. Nauk 29
(1974), no. 2(176), 11–49 (Russian). Collection of articles
dedicated to the memory of Ivan Georgievič Petrovskiĭ\
(1901–1973), I. MR 0516034
(58 #24324)
 [2]
Lawrence
M. Graves, The theory of functions of real variables,
McGrawHill Book Company, Inc., New YorkTorontoLondon, 1956. 2d ed. MR 0075256
(17,717f)
 [3]
John
Milnor, Singular points of complex hypersurfaces, Annals of
Mathematics Studies, No. 61, Princeton University Press, Princeton, N.J.;
University of Tokyo Press, Tokyo, 1968. MR 0239612
(39 #969)
 [4]
John
Milnor and Peter
Orlik, Isolated singularities defined by weighted homogeneous
polynomials, Topology 9 (1970), 385–393. MR 0293680
(45 #2757)
 [5]
Mutsuo
Oka, On the homotopy types of hypersurfaces defined by weighted
homogeneous polynomials, Topology 12 (1973),
19–32. MR
0309950 (46 #9053)
 [6]
Peter
Orlik, On the homology of weighted homogeneous manifolds,
Proceedings of the Second Conference on Compact Transformation Groups
(Univ. Massachusetts, Amherst, Mass., 1971) Springer, Berlin, 1972,
pp. 260–269. Lecture Notes in Math., Vol. 298. MR 0430307
(55 #3312)
 [7]
Peter
Orlik, Singularities and group
actions, Bull. Amer. Math. Soc. (N.S.)
1 (1979), no. 5,
703–720. MR
537624 (80e:57052), http://dx.doi.org/10.1090/S027309791979146433
 [8]
P.
Orlik and R.
Randell, The monodromy of weighted homogeneous singularities,
Invent. Math. 39 (1977), no. 3, 199–211. MR 0460320
(57 #314)
 [9]
Peter
Orlik and Philip
Wagreich, Isolated singularities of algebraic surfaces with C*\
action, Ann. of Math. (2) 93 (1971), 205–228.
MR
0284435 (44 #1662)
 [10]
, Algebraic surfaces with action, Acta. Math. 138 (1977), 4381.
 [1]
 V. I. Arnold, Normal forms of functions in neighborhoods of degenerate critical points, Russian Math. Surveys 29 (1974), 1050. MR 0516034 (58:24324)
 [2]
 Lawrence Graves, The theory of functions of real variables, 2nd ed., McGrawHill, New York, 1956. MR 0075256 (17:717f)
 [3]
 John Milnor, Singular points of complex hypersurfaces, Ann. of Math. Studies, no. 61, Princeton Univ. Press, Princeton, N. J., 1963. MR 0239612 (39:969)
 [4]
 John Milnor and Peter Orlik, Isolated singularities defined by weighted homogeneous polynomials, Topology 9 (1970), 385392. MR 0293680 (45:2757)
 [5]
 Mutsuo Oka, On the homotopy types of hypersurfaces defined by weighted homogeneous polynomials, Topology 12 (1973), 1932. MR 0309950 (46:9053)
 [6]
 Peter Orlik, On the homology of weighted homogeneous manifolds, Proc. Second Conf. Transformation Groups. I, Lecture Notes in Math., vol. 298, SpringerVerlag, Berlin, 1972, pp. 260269. MR 0430307 (55:3312)
 [7]
 , Singularities and group actions, Bull. Amer. Math. Soc. (N.S.) 1 (1979), 703720. MR 537624 (80e:57052)
 [8]
 Peter Orlik and Richard Randell, The monodromy of weighted homogeneous singularities, Invent. Math. 39 (1977), 199211. MR 0460320 (57:314)
 [9]
 Peter Orlik and Philip Wagreich, Isolated singularities of algebraic surfaces with action, Ann. of Math. (2) 93 (1971), 205228. MR 0284435 (44:1662)
 [10]
 , Algebraic surfaces with action, Acta. Math. 138 (1977), 4381.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S0002994719820637033X
PII:
S 00029947(1982)0637033X
Keywords:
Isolated singularity,
weighted homogeneous polynomial,
Milnor fibre,
monodromy
Article copyright:
© Copyright 1982
American Mathematical Society
