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Quadratic forms and holomorphic foliations on singular surfaces. (English) Zbl 0657.32007

Let \(V^ 2\) be a complex normal irreducible surface and \({\mathcal F}_ p\) a germ of holomorphic foliation singular at \(p\in V^ 2\). The author shows that if the resolution graph of \(V^ 2\) at the point p is a tree, then \({\mathcal F}_ p\) admits an invariant analytic curve through p. This is a generalization of the main theorem of an earlier paper of the author and P. Sad [Ann. Math., II. Ser. 115, 579-595 (1982; Zbl 0503.32007)] in which the result is established for \(V^ 2={\mathbb{C}}^ 2.\)
The proof is based on the technique of the resolution of singular foliations and some results about weighted graphs and their quadratic forms. Finally, the author shows the existence of germs of holomorphic foliations with no invariant curves through singular points of certain normal surfaces whose resolution graphs are cyclic.
Reviewer: A.Némethi

MSC:

32B99 Local analytic geometry

Citations:

Zbl 0503.32007
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References:

[1] Bendixson, I.: Sur les points singuliers des équations differentielles. Ofv. Kangl. Vetenskaps. Akade. Förhandlinger, (Stokholm)9, 635-658 (1898) · JFM 29.0275.01
[2] Briot, C.A., Bouquet, J.C.: J. Éc. Polyt. Cah.36, 133 (1856)
[3] Camacho, C., Sad, P.: Invariant varieties through singularities of holomorphic vector fields Ann. Math.115, 579-595 (1982) · Zbl 0503.32007 · doi:10.2307/2007013
[4] Du Val, P.: On absolute and non-absolute singularities of algebraic surfaces. Rev. Fac. Sci. Univ. Istanbul, Ser. A91, 159-215 (1944) · Zbl 0063.07941
[5] Gomez-Mont, X.: Foliations in complex analytic spaces. The Lefschets Centennial Conference Part III. Diff. Equations, Verjovsky, A. (ed.). Contemp. Math.58, 3, 127-139 (1986)
[6] Grauert, H.: Über Modifikationen und exzeptionelle analytische Menge. Math. Ann.146, 331-368 (1962) · Zbl 0173.33004 · doi:10.1007/BF01441136
[7] Gunning, R.C.: Lectures on complex analytic varieties. Finite analytic mappings. Math. Notes. Princeton: Princeton University Press 1974 · Zbl 0331.32005
[8] Gunning, R.C., Rossi, H.: Analytic functions of several complex variables. New York: Prentice Hall 1965 · Zbl 0141.08601
[9] Ince, E.L.: Ordinary differential euqtions, p. 297. New York: Dover 1926 · JFM 52.0462.01
[10] Laufer, H.: Normal two dimensional singularities. Ann. Math. Stud. No. 71 (1971) · Zbl 0245.32005
[11] Laufer, H.: Taut two dimensional singularities. Math. Ann.205, 131-164 (1973). · Zbl 0281.32010 · doi:10.1007/BF01350842
[12] Lins, N.A.: Construction of singular holomorphic vector fields and foliations in dimension two. J. Differ. Geom.26, 1-31 (1987) · Zbl 0625.57012
[13] Moussu, R.: Les conjectures de R. Thom sur les singularités de feuilletages holomorphes Preprint · Zbl 0772.57030
[14] Mattei, J.F., Moussu, R.: Holonomie et intégrales premières. Ann. Sci. Ec. Norm. Super, IV. Ser.13, 469-523 (1980) · Zbl 0458.32005
[15] Mumford, D.: The topology of normal singularities of an algebraic surface and a criterion for simplicity. Publ. Math. Inst. Hautes Etud. Sci.9, 5-22 (1961) · Zbl 0108.16801 · doi:10.1007/BF02698717
[16] Narasimhan, R.: Introduction to the theory of analytic spaces (Lect. Notes Math., Vol. 25) Berlin Heidelberg New York: Springer 1966 · Zbl 0168.06003
[17] Seidenberg, A.: Reduction of singularities of the differential equationAdY=EdX. Am. J. Math.90, 248-269 (1968) · Zbl 0159.33303 · doi:10.2307/2373435
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