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Question: Let $X$ be a non-singular algebraic surface of general type. Suppose that the canonical class $K_X$ is an integer multiple of another class $L$. Let $\Sigma_k$ be a smooth curve of genus $k$ that represents the class $L$ in $X$ (here I assume such $\Sigma_k$ exists). Is it true that $\pi_1(\Sigma_k)$ surjects into $\pi_1(X)$? Do you know any relationships between them?

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    $\begingroup$ By a result of Nori (Corr 2.4B in "Zariski's conjecture and related problems"), if $C^2>0$ for an irreducible curve $C$ on a surface $X$ then $\pi_1(C)$ surjects on $\pi_1(X)$. Since $K^2>0$ for a general type surface it follows that $\Sigma_k^2>0$ as well, in your question, hence the required surjection. $\endgroup$
    – Maharana
    Commented Aug 18, 2011 at 2:02

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Yes, $L$ is an ample line bundle and hence $\Sigma_k$ is a smooth ample divisor. The Lefschetz hyperplane theorem now implies that $\pi_1(\Sigma_k)\to\pi_1(X)$ is surjective.

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  • $\begingroup$ Thank you. Can I apply Lefschetz hyperplane theorem if it is not known $L$ can be represented by a complex submanifold? All I know is my $\Sigma_k$ is a symplectic submanifold of $X$. $\endgroup$
    – Thom
    Commented Aug 17, 2011 at 20:04
  • $\begingroup$ Good point. Above I assumed that $\Sigma$ is the zero-set of a global section of $L$. $\endgroup$
    – J.C. Ottem
    Commented Aug 17, 2011 at 20:31
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    $\begingroup$ Maybe this is trivial, but I don't see why $L$ is ample and not only big. $\endgroup$
    – Henri
    Commented Aug 17, 2011 at 20:35

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