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I browse in Phillip Griffiths' Slides on historical development of Hodge-theory and these include a sketch of the original approach with Lefschetz used to study complex surfaces in his famous ...

Let $H$ be a very ample linear system on a smooth compact complex surface $X$ whose Kodaira dimension is $\geq 0$. A general element of $H$ is smooth and has genus $\geq 2$. Let $L\subset H$ be a ...