Holomorphic dual vector bundle ($E_x^{\ast }={\mathrm{Hom}}_{\mathbb{C}}(E_x,\mathbb{C})$)

Dualize the fibres of a holomorphic vector bundle. If its transitions are $g_{ij}$, the dual transitions are $g_{ij}^{-T}$, which are holomorphic and preserve the evaluation pairing. These define the natural holomorphic structure on the smooth dual bundle.