Vector Bundles：differential geometry vs algebraic geometry

Question

I am in trouble about the vector bundle part in the Friedman's book: Algebraic Surfaces and Holomorphic Vector Bundle, I know what is a vector bundle(or a fibre bundle) in the differential geometry, but it seems that vector bundles in the algebraic geometry are something different, so who can tell me what is connection or difference between the two kinds of vector bundles, how to understand vector bundles in the algebraic geometry?

Answer 1

They're the same thing; it's just that sometimes in algebraic geometry instead of talking about a vector bundle $E \xrightarrow{\pi} X$ explicitly, we talk about the sheaf $\mathscr{E}$ that associates to an open set $U \subset X$ the set of all sections $s : U \to E$ of $X$, and call this the "vector bundle." The sheaves that arise in such a way are precisely the locally free sheaves on $X$.