Journal of Algebraic Geometry

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		Online ISSN 1534-7486; Print ISSN 1056-3911

The Multiplicity Polar Theorem and isolated singularities

Author(s): Terence Gaffney
Journal: J. Algebraic Geom. 18 (2009), 547-574.
Posted: November 17, 2008
MathSciNet review: 2496457
Retrieve article in: PDF

Abstract | References | Additional information

Abstract: We show how the Multiplicity Polar Theorem can be used to calculate invariants which describe an ``isolated singularity''. Examples include the defect of a function, which is related to the Euler obstruction, the index of a differential form, the dimension of the space of vanishing cycles of a sheaf of D-modules relative to a function at 0, and a formula for the relative cohomology of the Milnor fiber of where has an isolated singularity on a complex analytic set with possibly non-isolated singularities. We apply the result on the defect to refine previous work on the $\mathrm{A}_f$ condition.

References:

1.: J.-P. Brasselet, Existence des classes de Chern en théorie bivariante, Astérisque, vol. 101-102, 7-22, 1983. MR 737926 (85j:32019)
2.: J.-P. Brasselet, D. T. Lê, and J. Seade, Euler obstruction and indices of vector fields, Topology vol. 39, 1193-1208, 2000. MR 1783853 (2001i:32047)
3.: J.-P. Brasselet, D. Massey, A. J. Parameswaran and J. Seade, Euler Obstruction and Defects of Functions on Singular Varieties, J. London Math. Soc. (2) 70 (2004) 59-76. MR 2064752 (2005c:32037)
4.: J.-P. Brasselet and M.H. Schwartz, Sur les classes de Chern des ensembles analytiques complexes, Astérisque, vol. 82-83, 1981.
5.: W. Bruns and U. Vetter, Determinantal rings. Lecture Notes in Mathematics, 1327. Springer-Verlag, Berlin, 1988. MR 953963 (89i:13001)
6.: D. A. Buchsbaum and D. S. Rim, A generalized Koszul complex. II. Depth and multiplicity, Trans. Amer. Math. Soc. 111 (1963), 197-224. MR 0159860 (28:3076)
7.: W. Ebeling and S. M. Gusein-Zade, On the index of a vector field at an isolated singularity, The Arnoldfest, edited by E. Bierstone et al., Fields Inst. Commun. 24, AMS, 1999, pp. 141-152. MR 1733572 (2001k:32053)
8.: W. Ebeling and S. M. Gusein-Zade, On the index of a holomorphic 1-form on an isolated complete intersection singularity, Doklady Math. 64 (2001), 221-224. MR 1875501 (2002k:32059)
9.: W. Ebeling and S. M. Gusein-Zade, Indices of 1-forms on an isolated complete intersection singularity, Moscow Math. J. 3, 439-455 (2003). MR 2025268 (2005b:14007)
10.: W. Ebeling, S. M. Gusein-Zade, and J. Seade, Homological index for 1-forms and a Milnor number for isolated singularities, Internat. J. Math. 15 (2004) 895-905. MR 2106152 (2005i:32027)
11.: W. Fulton, Intersection Theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge $\cdot$ Band 2, Springer-Verlag, Berlin, 1984. MR 732620 (85k:14004)
12.: T. Gaffney, Aureoles and integral closure of modules, ``Stratifications, Singularities and Differential Equations II'', Travaux en Cours 55, Herman, Paris, 1997, 55-62. MR 1473241 (99a:32053)
13.: T. Gaffney, Polar Multiplicities and Equisingularity of Map Germs, Topology 32 (1993), 185-223. MR 1204414 (94f:32072)
14.: T. Gaffney, Integral closure of modules and Whitney equisingularity, Invent. Math. 107 (1992), 301-22. MR 1144426 (93d:32055)
15.: T. Gaffney, Plane sections, $\mathrm{W}_f$ and $\mathrm{A}_f$ , in ``Real and complex singularities (São Carlos, 1998)'', Chapman and Hall Res. Notes Math. 412, 2000, 17-32. MR 2088465 (2005d:32001)
16.: T. Gaffney, Multiplicities and equisingularity of ICIS germs, Invent. Math. 123 (1996), 209-220. MR 1374196 (97b:32051)
17.: T. Gaffney, Generalized Buchsbaum-Rim Multiplicities and a Theorem of Rees, Communications in Algebra, vol 31 #8 p3811-3828, 2003. MR 2007386 (2004m:13063)
18.: T. Gaffney, Polar methods, invariants of pairs of modules and equisingularity, Real and Complex Singularities (Sao Carlos, 2002), Ed. T.Gaffney and M.Ruas, Contemp. Math., #354, Amer. Math. Soc., Providence, RI, June 2004, 113-136. MR 2087808 (2005f:32044)
19.: T.Gaffney, The multiplicity of pairs of modules and hypersurface singularities, Real and Complex Singularities (Sao Carlos, 2004), Trends in Mathematics, Birkhäuser 2006, 143-168. MR 2280137 (2008e:32037)
20.: T. Gaffney, The Multiplicity-Polar Theorem, preprint 2007, math.CV/0703650. MR 2280137 (2008e:32037)
21.: T. Gaffney and S. Kleiman, Specialization of integral dependence for modules, Invent. Math. 137 (1999), 541-574. MR 1709870 (2000k:32025)
22.: M. Goresky and R. MacPherson, Stratified Morse Theory, Springer-Verlag, (1988). MR 932724 (90d:57039)
23.: G. M. Greuel, Der Gauss-Manin Zusammenhang isolierter Singularitäten von vollständigen Durchschnitten, Dissertation, Göttingen (1973), Math. Ann. 214 (1975), 235-266.MR 0396554 (53:417)
24.: S. Kleiman and A. Thorup, A geometric theory of the Buchsbaum-Rim multiplicity, J. Algebra 167 (1994), 168-231. MR 1282823 (96a:14007)
25.: S. Kleiman and A. Thorup, The exceptional fiber of a generalized conormal space, in ``New Developments in Singularity Theory.'' D. Siersma, C.T.C. Wall and V. Zakalyukin (eds.), Nato Science series, II Mathematics, Physics and Chemistry-Vol. 21 2001 401-404.
26.: D. T. Lê, Calculation of Milnor number of isolated singularity of complete intersection, Funct. Anal. 8 (1974), 127-31.
27.: D. T. Lê, Le concept de singularité isolée de fonction analytique, Advanced studies in pure math. 8 (1986), 215-227. MR 894295 (88d:32018)
28.: D. T. Lê, Morsification of D modules, Bol. Soc. Mat. Mexicana (3) 4 (1998), no. 2, 229-248. MR 1658264 (2000a:32063)
29.: R. MacPherson, Chern classes for singular varieties, Annals of Math, vol. 100, 423-432 1974. MR 0361141 (50:13587)
30.: D. Massey, Lê Cycles and Hypersurface Singularities, Springer Lecture Notes in Mathematics 1615, (1995). MR 1441075 (98h:32061)
31.: D. Massey, Hypercohomology of Milnor Fibers, Topology, v 35, #4, pp. 969-1003 (1996). MR 1404920 (97k:32055)
32.: D. Massey, Numerical Control over Complex Analytic Singularities, Memoirs of the AMS, #778, AMS 2003. MR 1962934 (2004d:32038)
33.: R. Pellikaan, Hypersurface singularities and resolutions of Jacobi modules, Thesis, Rijkuniversiteit Utrecht, (1985). MR 816348 (87j:32024)
34.: J. P. Serre, Local algebra. Translated from the French by CheeWhye Chin and revised by the author. Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2000. MR 1771925 (2001b:13001)
35.: B. Teissier, Multiplicités polaires, sections planes, et conditions de Whitney, in ``Proc. La Rábida, 1981.'' J. M. Aroca, R. Buchweitz, M. Giusti and M. Merle (eds.), Lecture Notes in Math. 961 (1982), 314-491. MR 708342 (85i:32019)

Additional Information:

Terence Gaffney
Affiliation: Department of Mathematics, 567 Lake Hall, Northeastern University, Boston, Massachusetts 02115
Address at time of publication: MSRI, 17 Gauss Way, Berkeley, California 94720-5070
Email: gaff@neu.edu

PII: S 1056-3911(08)00516-X
Received by editor(s): April 26, 2007
Received by editor(s) in revised form: January 11, 2008
Posted: November 17, 2008
Copyright of article: Copyright 2008, American Mathematical Society

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