  We investigate singular Hermitian metrics on vector bundles, especially
strictly Griffiths positive ones. $L^2$ esitimates and vanishing theorems
usually require an assumption that vector bundles are Nakano positive. However
there is no general definition of the Nakano positivity in singular settings.
In this paper, we show various $L^2$ estimates and vanishing theromes by
assuming that the vector bundle is strictly Griffiths positive and the base
manifold is projective.
