Holomorphic structures on associated bundles

Question

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!

Answer 1

The answer to both questions is yes. The first follows from Weyl's unitarian trick: $\pi$ can be extended to a holomorphic rep $\tilde\pi$ of $GL(n,\mathbb C)$. Then we can view $E$ as being associated to the principal $GL(n,\mathbb C)$ frame bundle of $M$ so $E$ must be holomorphic (it's transition functions are $\tilde \pi \circ g_{\alpha\beta}$ where $g_{\alpha\beta}$ are the transition functions for $TM$).

For the second answer if we work in a trivialization then $\nabla^M = d + A$ for some $A \in \Omega^1(M; \mathfrak{gl}(n,\mathbb C))$. Since this is the Chern connection, $A \in \Omega^{1,0}(M;\mathfrak{gl}(n,\mathbb C))$. Then the connection on $E$ will look like $d + \tilde \pi(A)$ which clearly has $(0,1)$ part equal to $\bar\partial$. Finally, that the connection on $E$ preserves the Hermitian structure is immediate since the representation $\pi$ is unitary.