Vanishing theorem for tame harmonic bundles via <i>L</i>²-cohomology

Using $L^2$ -methods, we prove a vanishing theorem for tame harmonic bundles over quasi-compact Kähler manifolds in a very general setting. As a special case, we give a completely new proof of the Kodaira-type vanishing theorems for Higgs bundles due to Arapura. To prove our vanishing theorem, we construct a fine resolution of the Dolbeault complex for tame harmonic bundles via the complex of sheaves of $L^2$ -forms, and we establish the Hörmander $L^2$ -estimate and solve $(\bar {\partial }_E+\theta )$ -equations for Higgs bundles $(E,\theta )$ .