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I'm reading the construction of difference bundle(generalized) in the paper of Atiyah-Hirzebruch(Analytic cycles on complex manifolds)

There is one thing I cannot understand. The followings are in the page 34 in the paper.

For exact sequence of vector bundles on $Y$.

$0 \to E_n \stackrel{a_n}{\to} E_{n-1} \to \dots \to E_1 \stackrel{a_1}{\to} E_0 \stackrel{a_0}{\to} 0$

Let $F_i := Ker(a_i)$. Then $F_i$ are vector bundles and the following short exact sequences

$0 \to F_i \to E_i \to F_{i-1} \to 0$ splits.

I cannot find the reason why they splits and there is no condition on the space $Y$. Is there a reason the above sequence splits, or the term 'split' used in a different sense?

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    $\begingroup$ If your base space is paracompact then it is possible to equip $E_i$ with a metric. Consider the decomposition of $E_i$ into $F_i$ and its orthogonal complement which is isomorphic to $F_{i-1}$. Now it is possible to construct a splitting map. $\endgroup$
    – Panagiotis Konstantis
    Commented Jan 25, 2019 at 13:16
  • $\begingroup$ I forgot to write something. In this case, $E_i$'s are complex vector bundles. Is it possible to induce complex structure to the orthogonal complement of $E_i$? It looks little strange for me because then, if the base space is paracompact, then every short exact sequence splits!. In general, it never happens in algebraic geometry. What is a difference when there is a metric? $\endgroup$
    – keaton
    Commented Jan 25, 2019 at 16:15
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    $\begingroup$ It is true that every short exact sequence of complex vector bundles split, by the above argument. However, it is not true that every short exact sequence of holomorphic vector bundles split, as in algebraic geometry. $\endgroup$
    – Tobias Shin
    Commented Jan 25, 2019 at 16:21
  • $\begingroup$ I think I understood what I missed. I appreciate for all comments :) $\endgroup$
    – keaton
    Commented Jan 25, 2019 at 16:40

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