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A remark on a non-vanishing theorem of P. Deligne and G. D. Mostow. (English) Zbl 0619.14006

Let X be an n-dimensional compact complex manifold and \(D=\sum D_ i\quad a\) normal crossing divisor on X. Let L be a local constant system of rank one on \(U=X-D\) none of whose monodromies around the \(D_ i's\) is one. Generalizing a criterion given by P. Deligne and G. D. Mostow in the case \(X={\mathbb{P}}^ 1\), we show that \(\omega \in H^ 0(U,\Omega^ n_ U\otimes_{{\mathbb{C}}}L)\) defines a non-vanishing cohomology class in \(H^ n(U,L)\), provided that \(\omega\) has a meromorphic extension to X and that for the closure Z of the zero divisor of \(\omega\) on U the sheaf \(\Omega^ n_ X(\log D)\otimes {\mathcal O}_ X(-Z)\) is numerically effective and of maximal Iitaka dimension.

MSC:

14C20 Divisors, linear systems, invertible sheaves
32L20 Vanishing theorems
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