MathOverflow will be down for maintenance for approximately 3 hours, starting Monday evening (06/24/2013) at approximately 9:00 PM Eastern time (UTC-4).

There is a general Lefschetz hyperplane theorem for the homotopy groups of an ample divisor $D$ on a projective smooth complex variety $X$. Basically, the this theorem says that the relative homotopy groups $\pi_i(X,D)$ are zero for all $i$ less than $n$. \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$. There is a very nice proof of this theorem using Morse theory and which can be found in Lazarsfeld's book 'Positivity in algebraic geometry'. 5 added 11 characters in body There is a general Lefschetz theorem for the homotopy groups of an ample divisor$D$on a projective variety$X$. Basically, the theorem says that the relative homotopy groups$\pi_i(X,D)$are zero for all$i$less than$n$. In particular, the map$\pi_1(D)\to\pi_1(X)$is an isomorphism for$\dim X\ge 3$and surjective for$\dim X\ge X= 2$. I think There is a very nice proof of this can be proved theorem using Morse theory and the proof which can be found in Lazarsfeld's book 'Positivity in algebraic geometry'. 4 added 1 characters in body There is a general Lefschetz theorem for the homotopy groups of an ample divisor$D$on a projective variety$X$. Basically, the theorem says that the relative homotopy groups$\pi_i(X,D)$are zero for all$i$less than$n$. In particular, the map$\pi_1(D)\to\pi_(X)$\pi_1(D)\to\pi_1(X)$ is an isomorphism for $\dim X\ge 3$ and surjective for $\dim X\ge 2$. I think this can be proved using Morse theory and the proof can be found in Lazarsfeld's book 'Positivity in algebraic geometry'.