4
$\begingroup$

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?

$\endgroup$
7
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.