  For a holomorphic vector bundle $E$ over a Hermitian manifold $M$ there are
two important notions of curvature positivity, the Griffiths positivity and
Nakano positivity. We study the consequence of these positivities and the
relevant estimates. If $E$ is Griffiths negative over K\"ahler manifold, then
there is a K\"ahler metric on its total space $E$, and we calculate the
curvature and prove the non-positivity of the curvature along the tautological
direction. The Nakano positivity can be formulated as a positivity for the
Nakano curvature operator and we give estimate the Nakano curvature operator
associated with a Nakano positive direct image bundle. As applications we
construct a mapping class group invariant K\"ahler metric on the quasi-Fuchsian
space QF$(S)$, which extends the Weil-Petersson metric on the Teichm\"uller
space $\mathcal{T}(S)\subset {\rm QF}(S)$, and we obtain estimates for the
Nakano curvature operator for the dual Weil-Petersson metric on the holomorphic
cotangent bundle of Teichm\"uller space.
