Hodge filtrations and the higher direct images of canonical sheaves
Let \(\pi\) : \(X\to S\) be a proper surjective morphism with connected fibers between nonsingular algebraic varieties over \({\mathbb{C}}\). Assume that there is an open subset \(S_ 0\) of S such that \(S\setminus S_ 0\) is a divisor with only normal crossings and that \(\pi_ 0:\quad X_ 0\to S_ 0\) is smooth, where \(X_ 0=\pi^{-1}(S_ 0)\) and \(\pi_ 0=\pi | X_ 0\). The local system \(R^ i\pi_{0*}{\mathbb{C}}_{X_ 0}\) on \(S_ 0\) forms a variation of Hodge structures with weight i for each \(i\geq 0\). Suppose that all the local monodromies of the local systems \(R^ i\pi_{0*}{\mathbb{C}}_{X_ 0}\) around \(S\setminus S_ 0\) are unipotent. Then there exists an isomorphism \(R^ i\pi_*\omega_{X/S}\cong F^ d({\mathcal H}_ S^{d+i})\) for all \(i\geq 0\), where \(d:=\dim X-\dim S\), \(\omega_{X/S}:=\omega_ X\otimes \pi^*\omega_ S^{-1}\) is the relative canonical sheaf, \({\mathcal H}^ i_ S\) is the canonical extension of \(R^ i\pi_{0*}{\mathbb{C}}_{X_ 0}\otimes_{{\mathbb{C}}_{S_ 0}}{\mathcal O}_{S_ 0}\) to S, and \(F^ d({\mathcal H}^ i_ S)\) is the d-th filtration of \({\mathcal H}^ i_ S\) induced from the d-th Hodge filtration.
scientific article; zbMATH DE number 3950693