Conjugate vector bundle ($\lambda \overline{v}=\overline{\overline{\lambda }v}$)

The conjugate bundle has the same underlying real bundle and the opposite complex scalar action. Its transitions are the complex conjugates of the original transitions. For a holomorphic bundle on $X$, these transitions are smooth and generally antiholomorphic on $X$; it is not automatically a holomorphic bundle there. It is naturally holomorphic over the conjugate complex manifold instead.

A complex bundle connection induces a connection on its conjugate vector bundle by the displayed rule on real tangent fields. The conjugate scalar action gives the correct Leibniz rule. In conjugate frames the matrix is the complex conjugate of the original connection matrix, with evaluation of one-forms on real tangent vectors understood.