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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?

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1 Answer 1

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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.

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  • $\begingroup$ Why do you need five? $\endgroup$
    – Mohan
    Jan 19, 2018 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. $\endgroup$
    – bvbrst
    Jan 19, 2018 at 1:34
  • $\begingroup$ @Mohan : it was a typo, fixed now. Thanks for your comment. $\endgroup$ Jan 19, 2018 at 6:11
  • $\begingroup$ @jhlee : exactly ! $\endgroup$ Jan 19, 2018 at 6:11
  • $\begingroup$ You should require $n\geq 3$, since plane curves are not in general simply connected, $\endgroup$
    – Nick L
    Nov 5, 2018 at 20:45

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