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- W2035996962 abstract "Let C be a complex affine reduced curve, and denote by H 1 ( C ) its first truncated cohomology group, i.e. the quotient of all regular differential 1-forms by exact 1-forms. First we introduce a nonnegative invariant μ ′ ( C , x ) that measures the complexity of the singularity of C at the point x , and we establish the following formula: dim H 1 ( C ) = dim H 1 ( C ) + ∑ x ∈ C μ ′ ( C , x ) where H 1 ( C ) denotes the first singular homology group of C with complex coefficients. Second we consider a family of curves given by the fibres of a dominant morphism f : X → C , where X is an irreducible complex affine surface. We analyze the behaviour of the function y ↦ dim H 1 ( f −1 ( y ) ) . More precisely we show that it is constant on a Zariski open set, and that it is lower semi-continuous in general. Soit C une courbe affine complexe réduite. Son premier groupe H 1 ( C ) de cohomologie tronqué est le quotient des 1-formes différentielles régulières sur C par les 1-formes régulières exactes. A tout point x de C , nous attachons un invariant positif μ ′ ( C , x ) qui mesure la complexité de la singularité ( C , x ) . Puis nous montrons la formule suivante : dim H 1 ( C ) = dim H 1 ( C ) + ∑ x ∈ C μ ′ ( C , x ) où H 1 ( C ) désigne le premier groupe d'homologie singulière de C à coefficients complexes. Ensuite nous considérons une famille de courbes données par les fibres d'un morphisme dominant f : X → C , où X est une surface affine complexe irréductible. Nous analysons le comportement de la fonction y → dim H 1 ( f −1 ( y ) ) . Plus précisément, nous montrons qu'elle est constante sur un ouvert de Zariski, et qu'elle est semi-continue inférieurement en général." @default.
- W2035996962 created "2016-06-24" @default.
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- W2035996962 date "2006-06-01" @default.
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- W2035996962 title "Cohomology of regular differential forms for affine curves" @default.
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- W2035996962 doi "https://doi.org/10.1016/j.bulsci.2005.11.002" @default.
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