Lokale topologische Eigenschaften komplexer Räume

Hamm, Helmut A.

doi:10.1007/bf01578709

Cited by 227 publications

(142 citation statements)

References 5 publications

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“…Dans [12], H. Hamm définit aussi un nombre de Milnor pour les intersections complètesà singularités isolées.…”

Section: Minimalité Du Nombre De Milnorunclassified

Limites d'espaces tangents � une surface normale

Snoussi¹

2001

Commentarii Mathematici Helvetici

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Résumé. Nousétudions l'ensemble des hyperplans limites d'hyperplans tangentsà un germe de surface normale. Nous caractérisons ces hyperplans par le fait que le nombre de Milnor de leur section avec la surface n'est pas minimum. Nous donnons ensuite une généralisation des résultats de [14] en termes de résolution simultanée faible de la famille des sections hyperplanes, ce qui nous permet de déterminer avec précision les tangentes exceptionnelles d'une surface normale. Grâceà ces résultats, nous démontrons que "les composantes de Tyurina" d'une désingularisation raisonnable se contractent sur des points fixes du système linéaire des courbes polaires. Abstract.We study the set of limiting tangent hyperplanes of a normal surface germ. We characterize these hyperplanes by the non-minimality of the Milnor number of their section with the surface. Then we generalise the results of [14] in terms of weak simultaneous resolution of the family of hyperplane sections, and hence we precisely determine the exceptional tangents of a normal surface singularity. Applying these results, we prove that the "Tyurina components" of a reasonable desingularisation contract to fixed points of the linear system of polar curves. Mathematics Subject Classification (2000). 32S05, 32S15, 32S25, 32S35, 32S45. Mots

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“…Dans [12], H. Hamm définit aussi un nombre de Milnor pour les intersections complètesà singularités isolées.…”

Section: Minimalité Du Nombre De Milnorunclassified

Limites d'espaces tangents � une surface normale

Snoussi¹

2001

Commentarii Mathematici Helvetici

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“…To Milnor [10], Hamm [5], the (i) is due. The assertion (ii) can be proved as follows: Take a smaller open ball B' and set X'=Xr\B'.…”

Section: Preliminariesmentioning

confidence: 99%

Some Remarks on Isolated Singularity and Their Application to Algebraic Manifolds

Naruki

1977

Publ. Res. Inst. Math. Sci.

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In 1954 F. Hirzebruch [8] obtained an interesting formula which makes it possible to determine the alternating sum X] 5 ( -z) 9 dim H q (V, J2 p (L fc )) for any complete intersection V of hypersurfaces in a complex projective space and for any k^Z where L is the analytic line bundle over V induced by hyperplanesection. He further determined dimH^y, J2 P ) by using some vanishing theorem. In the author's knowledge, however, the genera] dim H q (V, J2 P (Z/)) seem not to have been determined yet. In this note we shall give a formula which determines directly Part I is concerned with the general theory of isolated singularity and is of preparatory nature. The readers who have known the standard of the theory (e.g. Greuel [4]) may bypass it after they become familiar with the terminology and notations. Part II begins with the study of C*-actions over isolated singularities. The main theme there is to compute the characters of the representations of C* over various cohomology groups attached to the singularities, we apply it to the cones associated with algebraic manifolds and prove the required formula finally.The almost all results obtained in this paper have already been announced in [11], [12].Part I: General Theory I. PreliminariesWe shall often denote by (X, x) the pair of an analytic space X

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“…We assume now that (X, 0) has an isolated singularity, and we denote by f: CN -CP the mapping with components fl, . .. , fp; the affine manifold f '(y), where y is a regular value of f, is called the Milnor fibre of (X, 0) (see [13] and [8]). From [8] we thus know that f -1(y) has the homotopy type of a bouquet of n-dimensional spheres; the number of spheres in this bouquet is called the Milnor number of (X, 0), u(X, 0).…”

Section: Let Hd Be a Line Bundle Over Pn(c) Defined By The Linear Sysmentioning

confidence: 99%

“….. , fp; the affine manifold f '(y), where y is a regular value of f, is called the Milnor fibre of (X, 0) (see [13] and [8]). From [8] we thus know that f -1(y) has the homotopy type of a bouquet of n-dimensional spheres; the number of spheres in this bouquet is called the Milnor number of (X, 0), u(X, 0). The following result is originally stated in [7] (see also [6]).…”

Section: Let Hd Be a Line Bundle Over Pn(c) Defined By The Linear Sysmentioning

confidence: 99%

On the Chern classes and the Euler characteristic for nonsingular complete intersections

m.¹

1980

Proc. Amer. Math. Soc.

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It is shown that Hirzebruch’s result on the Chern classes of a complete intersection of nonsingular hypersurfaces in general position in P N ( C ) {{\mathbf {P}}_N}({\mathbf {C}}) , is also valid for any nonsingular complete intersection. Then relations between Euler characteristic, class and Milnor number are pointed out.

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Lokale topologische Eigenschaften komplexer Räume

Cited by 227 publications

References 5 publications

Limites d'espaces tangents � une surface normale

Limites d'espaces tangents � une surface normale

Some Remarks on Isolated Singularity and Their Application to Algebraic Manifolds

On the Chern classes and the Euler characteristic for nonsingular complete intersections

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