Torsion freeness of higher direct images of canonical bundles (Q1073856)

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.