A short analytic proof of closedness of logarithmic forms
Theorem: Let \(D\) be an effective reduced divisor on a compact Kähler manifold \(M\), let \(\omega\) be a logarithmic \(p\)-form over \(M\) along \(D\). Then \(d \omega = 0\) in \(M \backslash D\). Deligne proved it for \(M\) being complex projective by showing the degeneracy of a certain spectral sequence. In fact, his proof gives the result for a general \(M\). The author gives the theorem an analytic proof. He uses the desingularization theorem by Hironaka to assume that \(D\) has simple normal crossings only and then he proceeds by induction on \(p\) using the classical harmonic integral theory.
scientific article; zbMATH DE number 840320