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Higher direct images of log canonical divisors. (English) Zbl 1072.14019
Let $$f:X \to Y$$ be a surjective morphism of nonsingular projective varieties, and let $$D$$ be a simple normal crossing divisor on $$X$$ that is strongly horizontal with respect to $$f$$. Given a simple normal crossing divisor $$\Sigma$$ on $$Y$$, if $$f$$ is smooth and $$D$$ is relatively normal crossing over $$Y \setminus \Sigma$$, the author prove that $$R^i f_* \omega_{X/Y} (D)$$ is the upper canonical extension of the bottom Hodge filtration. This result is a logarithmic generalization of the theorem of Kollár and Nakayama. As a corollary, he also obtains a generalization of Fujita-Kawamata’s semipositivity theorem.

##### MSC:
 14E30 Minimal model program (Mori theory, extremal rays) 14C20 Divisors, linear systems, invertible sheaves 14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
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