Vanishing Theorem for tame harmonic bundles via $L^2$-cohomology
Résumé
Using $L^2$ -methods, we prove a vanishing theorem for tame harmonic bundles over quasicompact 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 $\bar{\partial}+\theta)$-equations for Higgs bundles $(E,\theta)$.
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