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

Abstract

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