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Vector fields, separatrices and Kato surfaces

Adolfo Guillot (2014)

Annales de l’institut Fourier

We prove that a singular complex surface that admits a complete holomorphic vector field that has no invariant curve through a singular point of the surface is obtained from a Kato surface by contracting some divisor (in particular, it is compact). We also prove that, in a singular Stein surface endowed with a complete holomorphic vector field, a singular point of the surface where the zeros of the vector field do not accumulate is either a quasihomogeneous or a cyclic quotient singularity. We give...

Volume and multiplicities of real analytic sets

Guillaume Valette (2005)

Annales Polonici Mathematici

We give criteria of finite determinacy for the volume and multiplicities. Given an analytic set described by {v = 0}, we prove that the log-analytic expansion of the volume of the intersection of the set and a "little ball" is determined by that of the set defined by the Taylor expansion of v up to a certain order if the mapping v has an isolated singularity at the origin. We also compare the cardinalities of finite fibers of projections restricted to such a set.

Weights in cohomology and the Eilenberg-Moore spectral sequence

Matthias Franz, Andrzej Weber (2005)

Annales de l’institut Fourier

We show that in the category of complex algebraic varieties, the Eilenberg–Moore spectral sequence can be endowed with a weight filtration. This implies that it degenerates if all spaces involved have pure cohomology. As application, we compute the rational cohomology of an algebraic G -variety X ( G being a connected algebraic group) in terms of its equivariant cohomology provided that H G * ( X ) is pure. This is the case, for example, if X is smooth and has only finitely many orbits. We work in the category...

Weil's formulae and multiplicity

Maria Frontczak, Andrzej Miodek (1991)

Annales Polonici Mathematici

The integral representation for the multiplicity of an isolated zero of a holomorphic mapping f : ( ℂ n , 0 ) → ( ℂ n , 0 ) by means of Weil’s formulae is obtained.

Whitney regularity and generic wings

V. Navarro Aznar, David J. A. Trotman (1981)

Annales de l'institut Fourier

Given adjacent subanalytic strata ( X , Y ) in R n verifying Kuo’s ratio test ( r ) (resp. Verdier’s ( w ) -regularity) we find an open dense subset of the codimension k C 1 submanifolds W (wings) containing Y such that ( X ∩ W , Y ) is generically Whitney ( b π ) -regular is exactly one more than the dimension...

Zeta functions and blow-Nash equivalence

Goulwen Fichou (2005)

Annales Polonici Mathematici

We propose a refinement of the notion of blow-Nash equivalence between Nash function germs, which has been introduced in [2] as an analog in the Nash setting of the blow-analytic equivalence defined by T.-C. Kuo [13]. The new definition is more natural and geometric. Moreover, this equivalence relation still does not admit moduli for a Nash family of isolated singularities. But though the zeta functions constructed in [2] are no longer invariants for this new relation, thanks to a Denef & Loeser...

μ -constant monodromy groups and marked singularities

Claus Hertling (2011)

Annales de l’institut Fourier

μ -constant families of holomorphic function germs with isolated singularities are considered from a global perspective. First, a monodromy group from all families which contain a fixed singularity is studied. It consists of automorphisms of the Milnor lattice which respect not only the intersection form, but also the Seifert form and the monodromy. We conjecture that it contains all such automorphisms, modulo ± id . Second, marked singularities are defined and global moduli spaces for right equivalence...

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