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'.
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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'. |
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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'. |
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