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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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