The cohomology of a variation of polarized Hodge structures over a quasi - compact Kähler manifold

Let (M,ω0) be a compact Kähler manifold of dimension n and D a divisor of M with at most normal crossing singularities . Let D = ∪i=1Di , where the Di are smooth divisors of M . Denote M \ D by M , called a quasi-compact Kähler manifold. j : M → M is the inclusion mapping. According to [3], one can construct a complete Kähler metric ω on M which is a Poincaré-like metric near the divisor and of finite volume. Let {M,HC = HZ ⊗ C,∇ = ∇1,0 + ∇0,1,F = {F}p=1,S} be a rational variation of polarized Hodge structures with weight m over M such that each F is a holomorphic subbundle of the local system HC and ∇1,0 satisfies the Griffiths’ infinitesimal period relation