This question was asked here, but I think there will not be an answer, so let me recopy the question on Mathoverflow.
In the book of S. Kobayashi, hyperbolic complex spaces, there is the lemma (3.A.3) on the page 153, section Royden's extension lemma, whose statement is
Every exact sequence of holomorphic vector bundles $0\to E'\to E\to E''\to 0$ on a Stein manifold splits.
The author's argument is that the obstruction to a splitting lies in $H^1(X,\mathrm{Hom}(E'',E'))$ which is zero since $X$ is a Stein manifold.
Could someone explain what obstruction means and why it lies in this cohomology group? Thank you very much.
Here is the standard procedure. Tensor with dual of $E''$ to get,
$$0\to E''^*\otimes E'\to E''^*\otimes E\to E''^*\otimes E''\to 0.$$
Applying global sections, we get a map $H^0(E''^*\otimes E'')\to H^1(E''^*\otimes E')$. The image of the natural 'identity' element in the first vector space gives an element in the second, called an obstruction. If this element is zero (as in the case of a Stein manifold, where $H^1=0$), then the identity element can be lifted to $H^0(E''^*\otimes E)$ and easy to check that this means your exact sequence splits.