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Let $E$ be a holomorphic vector bundle endowed with a singular Hermitianmetric $H$. In this paper, we develop the harmonic theory on $(E,H)$. Then weextend several canonical results of J. Koll\'{a}r and K. Takegoshi to thissituation. In the end, we generalise Nakano's vanishing theorem.

The harmonic theory on a vector bundle with singular Hermitian metrics and positivity