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Relative cohomology of non exceptional forms. (Cohomologie relative de formes non exceptionnelles.) (French) Zbl 0991.58001
This paper concerns the study of germs of holomorphic differential 1-forms at a point (say the origin) of the complex plane $${\mathbb C}^2$$. Such a form $$\eta$$ is said to be formally $$\omega$$-exact (resp. $$\omega$$-exact) if $$\eta=a\omega + dh$$ for some formal power series $$a,h$$ in two variables (resp. for some convergent power series $$a,h$$ in two variables).
Here $$\omega$$ is a holomorphic differential 1-form such that $$\omega\wedge d\omega=0$$. It is shown that if $$\omega$$ is non exceptional and $$\eta=a\omega + dh$$ for some formal power series $$a,h$$ then $$a,h$$ are in fact convergent power series.
The condition ‘non exceptional’ refers to a technical property of the exceptional divisor relative to the resolution of singularities of $$\omega$$.
It is also proved that the same result is true (i.e. formally $$\omega$$-exactness implies $$\omega$$-exactness) for dicritical forms $$\omega$$.
##### MSC:
 58A10 Differential forms in global analysis 32S45 Modifications; resolution of singularities (complex-analytic aspects)
##### Keywords:
holomorphic differential 1-form; dicritical forms
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##### References:
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