Let $X, Y$ be varieties over $\mathbb{C}$, and the topology I am talking about is the Eucliden topology.


I am not sure if the following two results are true, and where can I find the references:


(1) If $X,Y$ are birational then they have the same fundamental group.


(2) If $X$ is contractible (or maybe simply connected), then it only has trivial line bundle. 



topology on varieties
(1) is not correct unless you assume your varieties to be smooth and projective; for instance $\Bbb{P}^1$ with 2 points identified is not simply connected. In the smooth projective case it is true, see SGA 1.


In (2), I assume you are talking about *topological* classification of line bundles. They are classified by their first Chern class, so if $H^2(X,\mathbb{Z})$ is zero (in particular if $X$ is contractible), every line bundle is topologically trivial. But any smooth projective variety of dimension $>0$ has a nontrivial line bundle, and many of them are simply connected.


