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I am reading the book 'Differential geometry of complex vector bundles' by S. Kobayashi, and finding it quite difficult. I don't have a very strong background on differential geometry, and the author skips many details and assumes many things without even mentioning them, making his arguments hard to follow. Can someone suggest some other good book\online notes which cover more or less the same material (at least for chapter I),with more details?

For example, I would like to have a proof of the equivalence between flat connections and flat structures on a vector bundle, which is discussed in section 1.2

Then in section 1.3 he proves that a complex vector bundle over a complex manifold is holomorphic iff it has a connection whose (0,2)-component is same as $\bar{\partial}$. I couldn't follow any of these proof completely.

Any help would be appreciated.

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    It would be good idea for you to list the subjects you want, so that people who do not have the book can also give you useful information... – 2017-02-26
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    @MarianoSuárez-Álvarez: I've edited my question. Thanks for the suggestion. – 2017-02-26

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