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.

  • $\begingroup$ Why do you need five? $\endgroup$ – Mohan Jan 19 '18 at 1:22
  • $\begingroup$ I think it was a typo; the same argument works for my question. I understood this answer as following: use the Veronese embedding to identify the hypersurface in $\mathbb{P}^4$ with a hyperplane in $\mathbb{P}^{126}$, use the hyperplane theorem (homotopy version) there, and bring it back. $\endgroup$ – jhlee Jan 19 '18 at 1:34
  • $\begingroup$ @Mohan : it was a typo, fixed now. Thanks for your comment. $\endgroup$ – Nicolas Hemelsoet Jan 19 '18 at 6:11
  • $\begingroup$ @jhlee : exactly ! $\endgroup$ – Nicolas Hemelsoet Jan 19 '18 at 6:11
  • $\begingroup$ You should require $n\geq 3$, since plane curves are not in general simply connected, $\endgroup$ – Nick L Nov 5 '18 at 20:45

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