## The index of a holomorphic flow with an isolated singularity.(English)Zbl 0725.32012

The index of a holomorphic vector field Z on the germ of a hypersurface $${\mathcal V}$$ with an isolated singularity is defined. The index coincides with the Hopf index in the smooth case. Formulae for the index in terms of the ideals defining Z and $${\mathcal V}$$ are given. Topological invariance of the index and the Chern class as well as formulae relating global invariants of the Poincaré-Hopf type are proven.
Reviewer: X.Gómez-Mont

### MSC:

 32S05 Local complex singularities 32S25 Complex surface and hypersurface singularities 32S50 Topological aspects of complex singularities: Lefschetz theorems, topological classification, invariants
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### References:

 [1] Abraham, R., Robin, J.: Transversal mappings and flows. New York: Benjamin 1967 · Zbl 0171.44404 [2] Arnold, V., Gusein-Zade, S., Varchenko, A.: Singularities of differentiable maps. I. Basel Boston Stuttgart: Birkh?user 1985 · Zbl 1290.58001 [3] Camacho, C., Lins, A., Sad, P.: Topological invariants and equidesingularization for holomorphic vector fields. J. Differ. Geom.20, 143-174 (1984) · Zbl 0576.32020 [4] Camacho, C.: Quadratic forms and holomorphic foliations on singular surfaces. Math. Ann.282, 177-184 (1988) · Zbl 0657.32007 [5] Camacho, C., Cano, F., Sad, P.: Desingularization of absolutely isolated singularities. Invent. Math.98, 351-369 (1989) · Zbl 0689.32008 [6] Cossart, V., Giraud, J., Urbanz, U.: Resolution of surface singularities. (Lect. Notes Math. vol. 1101) Berlin Heidelberg New York: Springer 1984 [7] Douady, A.: Flatness and privilege. Enseign. Math. II. Ser.14, 47-74 (1968) · Zbl 0183.35102 [8] Edwards, R.: The solution of the 4-dimensional annulus conjecture. (After F. Quinn) In: Four manifolds theory Gordon, C., Kirby, R. (eds.) Contemp. Math.35, 211-264 (1984) [9] Fischer, G.: Complex analytic geometry. (Lect. Notes Math. vol. 538) Berlin Heidelberg New York: Springer 1976 · Zbl 0343.32002 [10] Friberg, B.: A topological proof of a theorem of Kneser. Proc. Am. Math. Soc.39, 421-426 (1973) · Zbl 0273.57017 [11] Fuller, F.: A relation between degree and linking number. In: Algebraic topology and geometry. Fox, R., Spencer, D., Tucker, A. (eds.) pp. 258-262. Princeton: Princeton Univ. Press 1957 · Zbl 0086.37301 [12] G?mez-Mont, X.: Universal families of foliations by curves. In: Singularites d’?quations differentielles. Cerveau, D., Moussu, R. (eds.) Asterisque150-151, 109-109 (1987) [13] G?mez-Mont, X., Luengo, I.: Universal germs of vector fields in ?3 without a separatrix. Invent. Math. (to appear) · Zbl 0774.32019 [14] Griffiths, P., Harris, J.: Principles of algebraic geometry. New York: Wiley 1978 · Zbl 0408.14001 [15] Hector, G., Hirsch, U.: Introduction to the geometry of foliations. Braunschweig Wiesbaden: Vieweg 1981 · Zbl 0486.57002 [16] Hirzebruch, F.: Topological methods in algebraic geometry. Berlin Heidelberg New York: Springer 1966 · Zbl 0138.42001 [17] Kirby, R.: Stable homeomorphisms and the annulus conjecture. Ann. Math.89, 575-582 (1969) · Zbl 0176.22004 [18] Kirby, R., Siebenmann, L.: Foundational essays on topological manifolds, smoothings, and triangulations. Ann. Math. Stud.88 (1977) · Zbl 0361.57004 [19] Kister, J.: Microbundles are fiberbundles. Ann. Math. II. Ser.80, 190-199 (1964) · Zbl 0131.20602 [20] Kneser, H.: Die Deformationss?tze der einfach zusammenh?ngenden Fl?chen. Math. Z.25, 362-372 (1926) · JFM 52.0573.01 [21] Milnor, J.: Topology from the differentiable viewpoint. Univ. Press of Virginia 1965 · Zbl 0136.20402 [22] Milnor, J.: Singular points of complex hypersurfaces. Ann. Maths. Stud.61 (1968) · Zbl 0184.48405 [23] Milnor, J.: On the 3-dimensional Brieskorn manifoldsM p,q,r?. In: Knots, groups, and 3-manifolds. Neuwirth, L.P. (ed.) Ann. Math. Stud.84, 175-225 (1975) [24] Nemytskii, V., Stepanov, V.: Qualitative theory of differential equations. Princeton: Princeton Univ. Press 1972 · Zbl 0089.29502 [25] Quinn, F.: The embedding theorem for towers. In: Four manifold theory. Gordon, C., Kirby, R. (eds.) Contemp. Math.35, 462-472 (1984) [26] Seade, J.A.: The index of a vector field on a complex surface with singularities. In: Proc. Lefschetz Centennial Conf. Verjovsky, A. (ed.) Contemp. Math.58, part III, 225-232 (1987) · Zbl 0614.32016 [27] Stern, R.: On topological and piecewise linear vector fields. Topology14, 257-269 (1975) · Zbl 0308.57003 [28] Thomas, E.: Fields of tangent 2-planes on even-dimensional manifolds. Ann. Math.86, 334-347 (1967) · Zbl 0168.21401
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