Complex vector bundle ($E\to X$)

A complex vector bundle is a vector bundle with complex-linear local trivializations and transition maps. A rank-$r$ complex bundle has an underlying oriented real bundle of rank $2r$, using the orientation of complex bases. Rank-one bundles are complex line bundles, and their higher-rank counterparts have integral Chern classes.

This is a smoothly varying positive-definite Hermitian form on each fiber, without requiring a holomorphic base or bundle. A partition of unity glues local positive metrics into a global one. Applying smooth Gram-Schmidt to local frames then makes the transitions unitary, reducing the structure group of a vector bundle to $U(k)$. For a line bundle its unit vectors form a principal bundle for the circle group.

For an even-rank real vector bundle $\xi$, a stable complex structure is a complex structure on $\xi \oplus {\mathbb{R}}^{2s}$, considered up to adding trivial complex bundles and homotopy. This gives a stable complex class but does not necessarily make $\xi$ itself a complex vector bundle. Stable Chern classes therefore need not recover the unstable Euler class: $T{S}^4\oplus {\mathbb{R}}^2$ can be made trivial complex, while $e(T{S}^4)$ evaluates to $2$.

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.