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Some consequences of perversity of vanishing cycles

Alexandru Dimca, Morihiko Saito (2004)

Annales de l’institut Fourier

For a holomorphic function on a complex manifold, we show that the vanishing cohomology of lower degree at a point is determined by that for the points near it, using the perversity of the vanishing cycle complex. We calculate this order of vanishing explicitly in the case the hypersurface has simple normal crossings outside the point. We also give some applications to the size of Jordan blocks for monodromy.

Sur certaines singularités non isolées d’hypersurfaces I

Daniel Barlet (2006)

Bulletin de la Société Mathématique de France

L’objectif de cet article est de mettre en place, dans le cadre de fonctions à lieu singulier de dimension 1, avec des hypothèses assez restrictives mais donnant accès à beaucoup d’exemples non triviaux, l’analogue de la théorie de E.Brieskorn pour une fonction à singularité isolée. Les principaux résultats sont le théorème de finitude pour le ( a , b ) -module associé à l’origine, qui est obtenu via le théorème de constructibilité de M. Kashiwara, et les résultats de non torsion pour une courbe plane (non...

Sur les fonctions à lieu singulier de dimension 1

Daniel Barlet (2009)

Bulletin de la Société Mathématique de France

Dans notre article [6] nous avons construit, pour une classe assez large de germes de fonctions holomorphes f : ( n + 1 , 0 ) ( , 0 ) à lieu singulier S : = { d f = 0 } de dimension 1 des invariants analytiques qui généralisent le réseau de Brieskorn d’un germe à singularité isolée. Dans cet article nous montrons que les résultats que nous avions obtenus s’étendent àtous les germes à lieu singulier de dimension 1 sans autre restriction. Ces invariants, essentiellement donnés par des (a,b)-modules géométriques, (objet qui est une abstraction...

The end curve theorem for normal complex surface singularities

Walter D. Neumann, Jonathan Wahl (2010)

Journal of the European Mathematical Society

We prove the “End Curve Theorem,” which states that a normal surface singularity ( X , o ) with rational homology sphere link Σ is a splice quotient singularity if and only if it has an end curve function for each leaf of a good resolution tree. An “end curve function” is an analytic function ( X , o ) ( , 0 ) whose zero set intersects Σ in the knot given by a meridian curve of the exceptional curve corresponding to the given leaf. A “splice quotient singularity” ( X , o ) is described by giving an explicit set of equations describing...

The Euler number of the normalization of an algebraic threefold with ordinary singularities

Shoji Tsuboi (2004)

Banach Center Publications

By a classical formula due to Enriques, the Euler number χ(X) of the non-singular normalization X of an algebraic surface S with ordinary singularities in P³(ℂ) is given by χ(X) = n(n²-4n+6) - (3n-8)m + 3t - 2γ, where n is the degree of S, m the degree of the double curve (singular locus) D S of S, t is the cardinal number of the triple points of S, and γ the cardinal number of the cuspidal points of S. In this article we shall give a similar formula for an algebraic threefold with ordinary singularities...

The geometric genus of hypersurface singularities

András Némethi, Baldur Sigurdsson (2016)

Journal of the European Mathematical Society

Using the path lattice cohomology we provide a conceptual topological characterization of the geometric genus for certain complex normal surface singularities with rational homology sphere links, which is uniformly valid for all superisolated and Newton non-degenerate hypersurface singularities.

The index of a vector field tangent to a hypersurface and the signature of the relative jacobian determinant

Xavier Gómez-Mont, Pavao Mardešić (1997)

Annales de l'institut Fourier

Given a real analytic vector field tangent to a hypersurface V with an algebraically isolated singularity we introduce a relative Jacobian determinant in the finite dimensional algebra B Ann B ( h ) associated with the singularity of the vector field on V . We show that the relative Jacobian generates a 1-dimensional non-zero minimal ideal. With its help we introduce a non-degenerate bilinear pairing, and its signature measures the size of this point with sign. The signature satisfies a law of conservation of...

The Milnor number of functions on singular hypersurfaces

Mariusz Zając (1996)

Banach Center Publications

The behaviour of a holomorphic map germ at a critical point has always been an important part of the singularity theory. It is generally known (cf. [5]) that we can associate an integer invariant - called the multiplicity - to each isolated critical point of a holomorphic function of many variables. Several years later it was noticed that similar invariants exist for function germs defined on isolated hypersurface singularities (see [1]). The present paper aims to show a simple approach to critical...

The Seiberg–Witten invariants of negative definite plumbed 3-manifolds

András Némethi (2011)

Journal of the European Mathematical Society

Assume that Γ is a connected negative definite plumbing graph, and that the associated plumbed 3-manifold M is a rational homology sphere. We provide two new combinatorial formulae for the Seiberg–Witten invariant of M . The first one is the constant term of a ‘multivariable Hilbert polynomial’, it reflects in a conceptual way the structure of the graph Γ , and emphasizes the subtle parallelism between these topological invariants and the analytic invariants of normal surface singularities. The second...

Toric embedded resolutions of quasi-ordinary hypersurface singularities

Pedro D. González Pérez (2003)

Annales de l’institut Fourier

We build two embedded resolution procedures of a quasi-ordinary singularity of complex analytic hypersurface, by using toric morphisms which depend only on the characteristic monomials associated to a quasi-ordinary projection of the singularity. This result answers an open problem of Lipman in Equisingularity and simultaneous resolution of singularities, Resolution of Singularities, Progress in Mathematics No. 181, 2000, 485- 503. In the first procedure the singularity is...

Un théorème à la « Thom-Sebastiani » pour les intégrales-fibres

Daniel Barlet (2010)

Annales de l’institut Fourier

L’objet de cet article est de démontrer un théorème «  à la Thom-Sebastiani  » pour les développements asymptotiques des intégrales-fibres des fonctions du type f g : ( x , y ) f ( x ) + g ( y ) sur ( p × q , ( 0 , 0 ) ) en terme des développements asymptotiques des intégrales-fibres associées aux germes holomorphes f : ( p , 0 ) ( , 0 ) et g : ( q , 0 ) ( , 0 ) . Ceci se ramène à calculer les développements asymptotiques d’une convolution Φ * Ψ à partir des développements asymptotiques de Φ et Ψ modulo les termes non singuliers.Pour obtenir un résultat précis donnant la non nullité des termes...

Une caractérisation des surfaces d'Inoue-Hirzebruch

Karl Oeljeklaus, Matei Toma, Dan Zaffran (2001)

Annales de l’institut Fourier

On montre que parmi les surfaces compactes complexes de classe V I I 0 avec b 2 > 0 , les surfaces d’Inoue-Hirzebruch sont caractérisées par le fait qu’elles possèdent deux champs de vecteurs tordus. Ce résultat est un pas vers la compréhension des feuilletages sur les surfaces V I I 0 .

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