Suppose $M$ is Kähler. Let $P \to M$ be the principal $U(n)$-frame bundle of $M$. Let $(\pi,V)$ be a finite dimensional unitary representation of $U(n)$ and let $E = P \times_\pi V$ be the associated complex vector bundle. I'm curious about answers and references to the following questions:
- Is $E$ naturally a holomorphic vector bundle?
- If so, is the connection on $E$ inherited by the Chern = Levi Civita connection on $M$, the Chern connection? Here I'm using the hermitian metric on $E$ induced by the hermitian inner product on $V$ and by Chern connection I mean the unique connection on $E$ compatible with the hermitian metric and such that $\nabla^{0,1} = \bar\partial_E$, the holomorphic structure on $E$.
Thanks!