I saw the following statement made without proof in a paper of Bogomolov and Tschinkel:


If $X$ is an algebraic surface, and $C$ is an ample smooth curve in $X,$ then the fundamental group of $C$ surjects onto that of $X.$


I was wondering if someone could supply a reference, and perhaps some idea of what the most general version of this statement was...



fundamental groups of curves
There is a general Lefschetz hyperplane theorem for the homotopy groups of an ample divisor $D$ on a smooth complex variety $X$. Basically, this theorem says that the relative homotopy groups $\pi\_i(X,D)$ are zero for all $i$ less than $\dim X$. In particular, the map $\pi\_1(D)\to\pi\_1(X)$ is an isomorphism for $\dim X\ge 3$ and surjective for $\dim X= 2$. There is a very nice proof of this theorem using Morse theory which can be found in Lazarsfeld's book 'Positivity in algebraic geometry'.


