I am reading some notes on mirror symmetry that discusses Calabi-Yau manifolds, which are defined as compact, connected, simply connected Kähler manifold whose canonical bundle is trivial. The main example of a Calabi-Yau (and mirror pairs) is given by a smooth quintic in $\mathbb{P}^4$, i.e. a zero set of a degree $5$ homogeneous polynomial.
However, I am having trouble understanding why this example must be a Calabi-Yau in our definition. Compactness seems okay, and it is Kähler as a smooth complex projective variety. Its first Chern class can be shown to be zero so the Calabi-Yau condition seems to hold too.
What bugs me is the simply connected assumption. For the notes do not provide an explanation, I guess that it must be some easy argument or a general, well-known fact that such a variety must be simply connected. Can someone provide such an argument or a reference?
If $n \geq 3$ and $X \subset \Bbb P^n$ is a smooth hypersurface, you can apply Lefschetz hyperplane theorem for deduce that $X$ is simply connected.