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I am a bit confused about the following fact ( cf Griffiths and Harris, principles of algebraic geometry):

Let $E$ be a hermitian vector bundle on a complex manifold M.

Let $D$ be a connexion on E such that $D^{0,1}=\overline{\partial}$ and $D$ compatible wth the metric. The fact that D is compatible with the metric implies that $D_{ij}$ ( the matrix associated to D in a unitary frame) is skew hermitian. But when I do the computations and using the fact that $D^{0,1}=\overline{\partial}$, I find that the $D_{ij}$ are necessarily (1,0) forms. and so are nul because the matrix is supposed to be skew-hermitian. If someone can tell me where I am making a confusion, it would be great. Thanks by advance.

Axel S
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  • I think I found the problem : The compatibilty with the complex structure does not imply the fact that $D_{ij}$ are (1,0) forms. The mistake I made was to considere the frame as an holomorphic one and so I forgot to apply the $\overline{\partial}$ operator to the frame. – Axel S Jan 21 '18 at 01:42

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