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Torsion freeness of higher direct images of canonical bundles. (English) Zbl 0589.14016
Let X be a complex manifold, Y a reduced and irreducible complex space and let $$f:\quad X\to Y$$ be a projective morphism. Assume that every connected component of X is mapped surjectively to Y. Then, the main purpose of this paper is to prove that $$R^ if_*\omega_ X$$ is torsion free for every $$i\geq 0$$, which is a local-analytic version of Kollár’s theorem, where $$\omega_ X$$ is the canonical bundle of X. We assume furthermore that Y is non-singular, X is equidimensional and that there is an effective divisor D on Y with only normal crossings such that f is smooth over $$Y\setminus D$$. Set $${\mathcal H}^ j:=R^ jf_*({\mathbb{Z}})|_{Y\setminus D}\otimes {\mathcal O}_{Y\setminus D}$$ and let $$F^ p({\mathcal H}^ j)$$ be the p-th Hodge filtration of $${\mathcal H}^ j$$. We set $$d=\dim X-\dim Y$$. Then we can show that there are isomorphisms $$\phi_ i:\quad R^ if_*\omega_{X/Y}\simeq F^ d(^ u{\mathcal H}^{d+i})$$ and $$\psi_ i:\quad R^ if_*{\mathcal O}_ X\simeq Gr^ 0_ F(^{\ell}{\mathcal H}^ i)$$ for every $$i\geq 0$$, where $$\omega_{X/Y}=\omega_ X\otimes f^*(\omega_ Y^{-1})$$ and $${}^ u{\mathcal H}^ j$$ (resp. $$^{\ell}{\mathcal H}^ j)$$ is the upper (resp. lower) canonical extension of the variation of Hodge structure $${\mathcal H}^ j$$. In particular, $$R^ if_*\omega_ X$$ and $$R^ if_*{\mathcal O}_ X$$ are locally free in this case. Moreover, we have that $$R^ if_*\omega_{X/Y}$$ is semi-positive and $$R^ if_*{\mathcal O}_ X$$ is semi-negative for every $$i\geq 0$$ if the local monodromies are unipotent.

##### MSC:
 14F05 Sheaves, derived categories of sheaves, etc. (MSC2010) 32L20 Vanishing theorems 14C30 Transcendental methods, Hodge theory (algebro-geometric aspects) 32L05 Holomorphic bundles and generalizations 14C20 Divisors, linear systems, invertible sheaves
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