I've read several different versions of Lefschetz hyperplane theorem. The statements are all saying the isomorphism of natural maps between (co)homology groups of degree $<n$, but have minor differences in the setting, i.e. :
- (Version 1) $Y$ be a projective of dimension $n$, $X$ be a very ample divisor such that $Y\backslash X$ smooth.
- (Version 2) $Y$ be smooth projective of dimension $n$, $X$ be a smooth very ample divisor.
- (Version 3) $Y$ be smooth projective of dimension $n$, $X$ be a smooth ample divisor.
Obviously (1) implies (2) and (3) implies (2). I think they should be essentially the same. But I don't know how to see (2) implies (1) and (3)?