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Vanishing theorem for tame harmonic bundles via L2-cohomology
- Part of
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- Journal:
- Compositio Mathematica / Volume 160 / Issue 12 / December 2024
- Published online by Cambridge University Press:
- 03 April 2025, pp. 2828-2855
- Print publication:
- December 2024
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- Article
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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 
$(E,\theta )$.
