# Metric connection on a holomorphic vector bundle

Hello!

Let $E\rightarrow M$ be a holomorphic vector bundle. We denote by $\mathcal{E}$ its sheaf of holomorphic sections and by $\mathcal{O}$ the sheaf of holomorphic functions on $M$. We also denote by $\nabla$ the unique flat $T^{1,0}M$-connection on $E$ (it seems that this is a standard fact from complex geometry that such a connection always exists on a holomorphic vector bundle). Let $\langle\cdot,\cdot\rangle$ be a smoothly varying $\mathbb{C}$-valued symmetric and non degenerate $\mathbb{C}$-bilinear form on the fibers on $E$ (this give us a metric).

I would like to show that this metric induces a homomorphism of sheaves of $\mathcal{O}$-modules $\mathcal{E}\otimes_{\mathcal{O}}\mathcal{E}\rightarrow\mathcal{O}$ (that is the metric becomes $\mathcal{O}$-bilinear) is equivalent to the fact that the connection $\nabla$ is metric, that is: $$X\cdot\langle\phi,\psi\rangle=\langle\nabla_X\phi,\psi\rangle+\langle\phi,\nabla_X\psi\rangle$$ for any smooth sections $\phi$ and $\psi$ of $E$ and any anti-holomorphic vector field $X$ on $M$ (I mean a section of $T^{0,1}M$).

Note that here, we do not assume that the metric is hermitian. This is a lemma stated without proof in a article I am reading. Since it comes without proof it is certainly a standard or simple fact. Can somebody give me a hint or a reference?

Please tell me if I am saying something wrong too.

Thank you!

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I'm probably misunderstanding something, but I'm not sure what you mean by your '$T^{1,0}M$-connection' $\nabla$. From your discussion further on, I suspect that you mean the operator ${\bar\partial}:C^\infty(E)\to C^\infty(E)\otimes\Omega^{0,1}(M)$, which is canonical, but isn't a 'connection', in the usual sense of the word. In any case, using $\bar\partial$ in the obvious way, your desired equation would be true if and only if the complex inner product were holomorphic, as you suspect. – Robert Bryant May 8 '11 at 13:33
@Robert Bryant: Thank you very much for your answer. You can see the lemma I was thinking about in the article "Holomorphic Poisson Manifolds and Holomorphic Lie Algebroids" available on arxiv.org from Laurent-Gengoux-Sténion-Xu, page 19, in which you have the definition of a $T^{1,0}M$-connection too. I am sorry because I thought this was something standard but it's not. – Benjamin May 8 '11 at 15:21