On the extension of twisted holomorphic sections of singular hermitian line bundles

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We prove a sharp Ohsawa-Takegoshi-Manivel type extension result for twisted holomorphic sections of singular hermitian line bundles over almost Stein manifolds. We establish as corollaries some extension results for pluri-twisted holomorphic sections of singular hermitian line bundles over projective manifolds.
Improvements of the results in section 4. Rest unchanged

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