Relative cohomology of non exceptional forms (Q1576408)

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 \(\omega\).