Equisingularity of families of hypersurfaces and applications to mappings. (English) Zbl 1233.32017
A family of isolated singularities is called Whitney equisingular if the singular set of the variety defined by the family is a stratum in a Whitney stratification. Briancon, Speder and Teissier proved that a family is Whitney equisingular if and only if the \(\mu^\ast\)-sequence is constant in the family, here \(\mu^\ast\) is the sequence of the Milnor numbers of generic \(i\)-dimensional plane sections with the singularity. In the case of non-isolated singularities one assumes that the family can be stratified in a way that outside the parameter axis of the family one has a Whitney stratification and seeks conditions that give an equivalence between a collection of topological invariants and Whitney equisingularity of the parameter axis.
In their paper [J. Algebr. Geom. 8, No. 4, 695–736 (1999; Zbl 0971.13021)], T. Gaffney and R. Gassler define the sequence \(\chi^\ast\) of Euler characteristics of the Milnor fibres generalizing \(\mu^\ast\). They define another sequence \(m^\ast\), the relative polar multiplicities, and prove that Whitney equisingularity of a family implies that \((m^\ast, \chi^\ast)\) are constant in the family. The aim of the polar is to give further conditions to ensure the converse.
In their paper [J. Algebr. Geom. 8, No. 4, 695–736 (1999; Zbl 0971.13021)], T. Gaffney and R. Gassler define the sequence \(\chi^\ast\) of Euler characteristics of the Milnor fibres generalizing \(\mu^\ast\). They define another sequence \(m^\ast\), the relative polar multiplicities, and prove that Whitney equisingularity of a family implies that \((m^\ast, \chi^\ast)\) are constant in the family. The aim of the polar is to give further conditions to ensure the converse.
Reviewer: Gerhard Pfister (Kaiserslautern)
MSC:
| 32S15 | Equisingularity (topological and analytic) |
| 32S30 | Deformations of complex singularities; vanishing cycles |
| 32S60 | Stratifications; constructible sheaves; intersection cohomology (complex-analytic aspects) |
Citations:
Zbl 0971.13021References:
| [1] | K. Bekka, C-régularité et trivialité topologique, Singularity theory and its applications, Part I (Coventry, 1988/89), Lecture Notes in Math., 1462, pp. 42-62, Springer-Verlag, Berlin, 1991. · Zbl 0733.58003 · doi:10.1007/BFb0086373 |
| [2] | J. Fernández de Bobadilla, Answers to some equisingularity questions, Invent. Math. 161 (2005), 657-675. · Zbl 1082.32019 · doi:10.1007/s00222-005-0441-4 |
| [3] | J. Fernández de Bobadilla and M. Pe Pereira, Equisingularity at the normalisation, J. Topol. 1 (2008), 879-909. · Zbl 1160.32025 · doi:10.1112/jtopol/jtn024 |
| [4] | N. Bourbaki, Éléments de mathématique, Algèbre commutative, Masson, Paris, 1983. |
| [5] | J. Briançon, P. Maisonobe, and M. Merle, Localisation de systèmes différentiels, stratifications de Whitney et condition de Thom, Invent. Math. 117 (1994), 531-550. · Zbl 0920.32010 · doi:10.1007/BF01232255 |
| [6] | J. Briançon and J.-P. Speder, La trivialité topologique n’implique pas les conditions de Whitney, C. R. Acad. Sci. Paris Sér. A-B 280 (1975), A365-A367. · Zbl 0331.32010 |
| [7] | J. Damon, Higher multiplicities and almost free divisors and complete intersections, Mem. Amer. Math. Soc. 123 (1996). · Zbl 0867.32015 |
| [8] | A. du Plessis and C. T. C. Wall, The geometry of topological stability, London Math. Soc. Monogr. (N.S.), 9, Oxford Univ. Press, New York, 1995. · Zbl 0870.57001 |
| [9] | T. Gaffney, Polar multiplicities and equisingularity of map germs, Topology 32 (1993), 185-223. · Zbl 0790.57020 · doi:10.1016/0040-9383(93)90045-W |
| [10] | —, Polar methods, invariants of pairs of modules and equisingularity, Real and complex singularities (T. Gaffney, M. A. S. Ruas, eds.) Contemp. Math., 354, pp. 113-135, Amer. Math. Soc., Providence, RI, 2004. · Zbl 1072.32020 |
| [11] | —, Condition W and mapgerms of kernel rank 1, preprint, June 2006. |
| [12] | T. Gaffney and R. Gassler, Segre numbers and hypersurface singularities, J. Algebraic Geom. 8 (1999), 695-736. · Zbl 0971.13021 |
| [13] | T. Gaffney and D. B. Massey, Trends in equisingularity theory, Singularity theory (Liverpool, 1996), London Math. Soc. Lecture Note Ser., 263, pp. 207-248, Cambridge Univ. Press, Cambridge, 1999. · Zbl 0966.14010 |
| [14] | M. Goresky and R. MacPherson, Stratified Morse theory, Ergeb. Math. Grenzgeb. (3), 14, Springer-Verlag, Berlin, 1988. · Zbl 0639.14012 |
| [15] | K. Houston, On the topology of augmentations and concatenations of singularities, Manuscripta Math. 117 (2005), 383-405. · Zbl 1089.32021 · doi:10.1007/s00229-005-0552-7 |
| [16] | —, On equisingularity of families of maps \((\mathbb C^n,0)\to(\mathbb C^n+1,0),\) Real and complex singularities (São Carlos, 2004), pp. 201-208, Trends Math., Birkhäuser, Basel, 2007. |
| [17] | —, Disentanglements and Whitney equisingularity, Houston J. Math. 33 (2007), 663-681. · Zbl 1140.32017 |
| [18] | —, Stratification of unfoldings of corank 1 singularities, Quart. J. Math. Oxford Ser. (2) 61 (2010), 413-435. · Zbl 1225.58016 · doi:10.1093/qmath/hap012 |
| [19] | V. H. Jorge Pérez, Polar multiplicities and equisingularity of map germs from \(\mathbb C^3\) to \(\mathbb C^3,\) Houston J. Math. 29 (2003), 901-924. · Zbl 1155.58308 |
| [20] | —, Polar multiplicities and equisingularity of map germs from \(\mathbb C^3\) to \(\mathbb C^4,\) Real and complex singularities (D. Mond, M. José Saia, eds.), Lecture Notes in Pure and Appl. Math., 232, pp. 207-225, Dekker, New York, 2003. · Zbl 1078.58506 |
| [21] | V. H. Jorge Pérez and M. Saia, Euler obstruction polar multiplicities and equisingularity of map germs in \(\mathcal O(n,p),\) \(n<p,\) Internat. J. Math. 17 (2006), 887-903. · Zbl 1108.58029 · doi:10.1142/S0129167X06003783 |
| [22] | D. T. Lê, Sur les cycles évanouissants des espaces analytiques, C. R. Acad. Sci. Paris Sér. A-B 288 (1979), A283-A285. · Zbl 0471.32007 |
| [23] | D. T. Lê and B. Teissier, Variétés polaires locales et classes de Chern des variétés singulières, Ann. of Math. (2) 114 (1981), 457-491. · Zbl 0488.32004 · doi:10.2307/1971299 |
| [24] | —, Cycles évanescents, sections planes, et conditions de Whitney II, Singularities, Part 2 (Arcata, 1981), Proc. Sympos. Pure Math., 44, pp. 65-103, Amer. Math. Soc., Providence, RI, 1983. |
| [25] | D. B. Massey, Numerical invariants of perverse sheaves, Duke Math. J. 73 (1994), 307-369. · Zbl 0799.32033 · doi:10.1215/S0012-7094-94-07315-8 |
| [26] | —, Lê cycles and hypersurface singularities, Lecture Notes in Math., 1615, Springer-Verlag, Berlin, 1995. · Zbl 0835.32002 |
| [27] | —, Numerical control over complex analytic singularities, Mem. Amer. Math. Soc. 163 (2003). · Zbl 1025.32010 |
| [28] | J. N. Mather, Stability of \(C^\infty\) mappings. VI: The nice dimensions, Proceedings of Liverpool Singularities Symposium I (1969/70), Lecture Notes in Math., 192, pp. 207-253, Springer-Verlag, Berlin, 1971. |
| [29] | D. Mond, On the classification of germs of maps from \(\mathbb R^2\) to \(\mathbb R^3,\) Proc. London Math. Soc. (3) 50 (1985), 333-369. · Zbl 0557.58006 · doi:10.1112/plms/s3-50.2.333 |
| [30] | A. Parusiński, Limits of tangent spaces to fibres and the \(w_f\) condition, Duke Math. J. 72 (1993), 99-108. · Zbl 0798.32030 · doi:10.1215/S0012-7094-93-07205-5 |
| [31] | B. Teissier, Variétés polaires. II. Multiplicités polaires, sections planes, et conditions de Whitney, Algebraic geometry (La Rábida, 1981), Lecture Notes in Math., 961, pp. 314-491, Springer-Verlag, Berlin, 1982. · Zbl 0585.14008 |
| [32] | C. T. C. Wall, Finite determinacy of smooth map-germs, Bull. London Math. Soc. 13 (1981), 481-539. · Zbl 0451.58009 · doi:10.1112/blms/13.6.481 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
