Is a complex line bundle over a compact Riemann surface topologically trivial iff it is holomorphically trivial? If so, how does one demonstrate that, and if not, what is a counterexample? 



Is there a toplogically trivial line bundle over a compact Riemann surfaces that isn't holomorphically trivial?
There are holomorphic line bundles over a compact Riemann surface $X$ that are topologically trivial, yet not holomorphically trivial. To see this, note that smooth complex line bundles are classified by a complete invariant, called the degree. By contrast, we have the Picard group $Pic(X)$ of isomorphism classes of holomorphic line bundles on $X$.


One always has a surjective group morphism $Pic(X)\rightarrow\mathbb{Z}$, defined by taking degrees of holomorphic line bundles. In general (ie. for positive genus), this map is not an isomorphism. Its kernel therefore contains smoothly (hence topologically) trivial complex line bundles that are not holomorphically trivial. 


