Vanishing theorem for tame harmonic bundles via $L^2$-cohomology

Abstract

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 $(E,\theta )$.

30 pages, comments are welcome!

References in corpus (3)

• Asymptotic behaviour of tame harmonic bundles and an application to pure twistor $D$-modules
• Semistable Higgs bundles of small ranks are strongly Higgs semistable
• Logarithmic vanishing theorems on compact Kähler manifolds I