TITLE: The fundamental group of an algebraic surface
QUESTION [6 upvotes]: 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?
Thanks

REPLY [2 votes]: 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.