Higher direct images of log canonical divisors. (English) Zbl 1072.14019

Fujino, Osamu

J. Differ. Geom. 66, No. 3, 453-479 (2004).

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.

Reviewer: Min Ho Lee (Cedar Falls)

MSC:

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