Is $\bigwedge^{p,0}M$ a holomorphic vector bundle?

Question

I have difficulties in understanding how to show that a vector bundle is holomorphic. For instance, how can I prove that $\bigwedge^{p,0}M$ is a holomorphic vector bundle, where $M$ is a complex manifold ? I think I should determine the transition maps and show that they are holomorphic. However, it is unclear to me how to do this in a correct way.

Thanks in advance.

Answer 1

A complex rank r vector bundle $E$ over $M$ is given by a cocycle $$\{\varphi_{ij}\colon U_i\cap U_j \longrightarrow {\rm GL}(r,\mathbb{C}) \}.$$ $E$ is holomorphic if the $\varphi_{ij}$ are holomorphic. Then we can apply "any linear algebra construction" such as exterior and symmetric powers to $E$ (fiberwise) and the result will be also a holomorphic bundle. For example, the determinant of $E$ will have a cocycle given by $\det(\varphi_{ij})$.

See page 67 of Huybrecht's book https://www.math.uh.edu/~shanyuji/Complex/Complexgeometry.pdf