A general calculation of the number of vanishing cycles. (English) Zbl 0824.32013
Let \(X\) be a reduced complex analytic space containing an open neighborhood \(U\) of the origin in \(\mathbb{C}^ n\) and let \(f : (U,0) \to (\mathbb{C})\) be an analytic function. Suppose that \(\mathbb{F}^ \bullet\) is a bounded complex of sheaves on \(X\) which is constructible with respect to a Whitney stratification \(\{S_ \alpha\}\) of \(X\). Let \(H\) be a hypersurface in \(U\) which transversely intersects the strata of \(\{S_ \alpha\}\) except, perhaps, at the origin. The author obtains an algebraic formula for calculating the difference between the Euler characteristics of the stalks cohomology at the origin of the sheaves of vanishing cycles along \(f\) and along the restriction \(f \mid_ H\). Thus, he generalizes a result of Le Dung Trang [Ann. Inst. Fourier 23(1973), No. 4, 261- 270 (1974; Zbl 0293.32013)]. In the case where \(\mathbb{F}^ \bullet\) is a perverse sheaf on a complete intersection the formula (5.11a) of E. J. N. Looijenga [‘Isolated singular points on complete intersections’, Lond. Math. Soc. Lectures Note Series 77, Cambridge (1984; Zbl 0552.14002)] is also generalized. As an application it is proved that the complex link of any isolated complete intersection singularity is never contractible.
Reviewer: A.G.Aleksandrov (Moskva)
MSC:
| 32S60 | Stratifications; constructible sheaves; intersection cohomology (complex-analytic aspects) |
| 32S55 | Milnor fibration; relations with knot theory |
| 32S30 | Deformations of complex singularities; vanishing cycles |
| 14B05 | Singularities in algebraic geometry |
| 14F43 | Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies) |
| 57M25 | Knots and links in the \(3\)-sphere (MSC2010) |
| 57Q45 | Knots and links in high dimensions (PL-topology) (MSC2010) |
Keywords:
isolated singularities; polar curve; vanishing cycles; perverse shaves; constructible sheaves; characteristic cycle; Euler characteristic; complete intersections; complex link of a singularityReferences:
| [1] | A’Campo, N., Le nombre de Lefschetz d’une monodromie, (Proc. Kon. Nedel. Akad. Wetensch. Ser., A 76 (1973)), 113-118 · Zbl 0276.14004 |
| [2] | Beilinson, A.; Berstein, J.; Deligne, P., Faisceaux pervers, Astérisque, 100 (1983) · Zbl 0536.14011 |
| [3] | Briançon, J.; Maisonobe, P.; Merle, M., Localisation de systèmes différentiels, stratifications de Whitney et condition de Thom (1992), Preprint · Zbl 0920.32010 |
| [4] | Briançon, J.; Speder, J. P., La trivialisté topologique n’implique pas les conditions de Whitney, C.R. Acad. Sci. Paris Sér., A 280 (1975) · Zbl 0331.32010 |
| [5] | Brylinski, J., Transformations canoniques, dualité projective, théorie de Lefschetz, transformations de Fourier et sommes trigonométriques, Astérisque, 140 (1986) · Zbl 0624.32009 |
| [6] | Brylinski, J.; Dubson, A.; Kashiwara, M., Formule de l’indice pour les modules holonomes et obstruction d’Euler locale, C.R. Acad. Sci. Sér., A 293, 573-576 (1981) · Zbl 0492.58021 |
| [7] | Deligne, P., Comparaison avec la théorie transcendante, (Séminaire de Géométrie Algébrique du Bois-Marie. Séminaire de Géométrie Algébrique du Bois-Marie, Lecture Notes in Mathematics, 340 (1973), Springer: Springer Berlin), SGA 7 II · Zbl 0266.14009 |
| [8] | Fulton, W., Intersection Theory, (Ergebnisse der Mathematical und ihrer Grenzgebiele, 2 (1984), Springer: Springer Berlin), (3) · Zbl 0541.14005 |
| [9] | Ginsburg, V., Characteristic varieties and vanishing cycles, Invent. Math., 84, 327-403 (1986) · Zbl 0598.32013 |
| [10] | Goresky, M.; MacPherson, R., Morse theory and intersection homology, (Analyse et Topologie sur les Espaces Singuliers. Analyse et Topologie sur les Espaces Singuliers, Astérisque, 101 (1983)), 135-192 · Zbl 0524.57022 |
| [11] | Gorosky, M.; MacPherson, R., Intersection homology II, Invent. Math., 71, 77-129 (1983) · Zbl 0529.55007 |
| [12] | Goresky, M.; MacPherson, R., Stratified Morse Theory, (Ergebnisse der Mathematic und ihrer Grenzgebiele, 14 (1988), Springer: Springer Berlin), (3) · Zbl 0526.57022 |
| [13] | Hamm, H.; Lê, D. T., Un Theoreme de Zariski du type de Lefschetz, Ann. Sci. École Norm. Sup., 6, 317-366 (1973) · Zbl 0276.14003 |
| [14] | Hironaka, H., Stratification and flatness, (Real and Complex Singularities (1977), Nordic Summer School), 199-265, (Oslo, 1976) · Zbl 0424.32004 |
| [15] | Kashiwara, M.; Schapira, P., Sheaves on Manifolds, (Grundlehrer der Mathematischen Wissenschaften, 292 (1990), Springer: Springer New York) · Zbl 0709.18001 |
| [16] | Lê, D. T., Calcul du nombre de cycles evanouissants d’une hypersurface complexe, Ann. Inst. Fourier (Grenoble), 23, 261-270 (1973) · Zbl 0293.32013 |
| [17] | Lê, D. T., Topological use of polar curves, (Proceedings of Symposia in Pure Methametics, 29 (1975), American Mathematical Society: American Mathematical Society Providence, RI), 507-512 · Zbl 0323.14003 |
| [18] | Lê, D. T., The geometry of the monodromy theorem, (Ramanathan, K. G., C.P. Ramanujam, a tribute. C.P. Ramanujam, a tribute, Tata Inst. Stud. Math., 8 (1978)) · Zbl 0434.32010 |
| [19] | Lê, D. T., Sur les cycles évanouissants des espaces analytiques, C.R. Acad. Sci. Paris Sér., A 288, 283-285 (1979) · Zbl 0471.32007 |
| [20] | Lê, D. T., Morsification of D-modules (1988), Preprint |
| [21] | Lê, D. T., Complex analytic functions with isolated singularities (1991), Preprint |
| [22] | Lê, D. T.; Mebkhout, Z., Variétés caractéristiques et variétés polaires, C.R. Acad. Sci., 296, 132-1299 (1983) · Zbl 0566.32020 |
| [23] | Lê, D. T.; Teissier, B., Variétés polaires locales et classes de Chern des variétés singulières, Ann. of Math., 114, 457-491 (1981) · Zbl 0488.32004 |
| [24] | Lê, D. T.; Teissier, B., Cycles evanescents, sections planes et conditions de Whitney II, (Proceedings of Symposia Pure Mathematics, 40 (1983), American Mathematical Society: American Mathematical Society Providence, RI), 03-65, Part 2 · Zbl 0532.32003 |
| [25] | Looijenga, E. J.N., Isolated Singular Points on Complete Intersections, (London Mathematical Society Lecture Note Series, 77 (1984), Cambridge University Press: Cambridge University Press Cambridge) · Zbl 0552.14002 |
| [26] | MacPherson, R., Global questions in the topology of singular spaces, (Proceedings International Congress of Mathematics. Proceedings International Congress of Mathematics, Warsaw (1983)), 213-235 · Zbl 0612.57012 |
| [27] | Massey, D., The Lê varieties I, Invent. Math., 99, 357-376 (1990) · Zbl 0712.32020 |
| [28] | Massey, D., Local Morse inequalities and perverse sheaves (1990), Preprint |
| [29] | Massey, D., The Thom condition along a line, Duke Math. J., 60, 631-642 (1990) · Zbl 0709.32026 |
| [30] | Massey, D., The Lê varieties II, Invent. Math., 104, 113-148 (1991) · Zbl 0727.32015 |
| [31] | D. Massey, Numerical invariants of perverse sheaves, to appear.; D. Massey, Numerical invariants of perverse sheaves, to appear. · Zbl 0799.32033 |
| [32] | Mather, J., (Notes on topological stability (1970), Harvard University), unpublished notes |
| [33] | Milnor, J., Singular Points of Complex Hypersurfaces, (Annals of Mathematics Studies, 61 (1968), Princeton University Press: Princeton University Press Princeton, NJ) · Zbl 0184.48405 |
| [34] | Teissier, B., Varietes polaires II: Multiplicites polaires, sections planes, et conditions de Whitney, (Algebraic Geometry, Proceedings. Algebraic Geometry, Proceedings, La Rabida 1981. Algebraic Geometry, Proceedings. Algebraic Geometry, Proceedings, La Rabida 1981, Springer Lecture Notes in Mathematics, 961 (1982), Springer: Springer Berlin), 314-491 · Zbl 0585.14008 |
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.
