The kernel spectral sequence of vanishing cycles (Q1813602)

Kernel filtration is the increasing filtration \(K\) defined by \(K_ k=\ker N^{k+1}\) for \(k\geq -1\), where \(N\) is the logarithm of the unipotent part of the monodromy. Let \(\psi_ f\) denote Deligne's vanishing cycle factor. The following theorem is proved: Let \(f:X\to S\) be a proper morphism of an irreducible analytic space \(X\) onto an open disc \(S\). Assume there exists a proper bimeromorphic morphism \(\pi:\tilde X\to X\) with \(X\) a Kähler manifold. Let \(L\) be a local system underlying a polarizable variation of Hodge structure on a smooth Zariski-open subset \(U\) of \(X\). Then the kernel spectral sequence \[ E_ 1^{-k,j+k}=H^ j(X_ 0,Gr_ k^ K{^ p\psi_ f}IC_ X(L))\Rightarrow H^ j(X_ 0,^ p\psi_ fIC_ X(L)) \] degenerates at \(E_ 2\), and the induced filtration on the abutment is the kernel filtration.