5
$\begingroup$

Let M be a compact complex manifold. Then is it true that if the first Chern class of M is even, then M admits a spin structure?

$\endgroup$
12
$\begingroup$

Yes. An oriented real vector bundle is spin if and only if its second Stiefel-Whitney class vanishes. If $E_\mathbb{C}$ is a complex vector bundle and $E_\mathbb{R}$ is the underlying real bundle then the second Stiefel-Whitney class is given by $w_2(E_\mathbb{R}) = c_1(E_\mathbb{C})$ mod 2. The details appear somewhere in chapter 2 of Spin Geometry by Lawson and Michelsohn.

$\endgroup$
  • 3
    $\begingroup$ More generally, the total Stiefel-Whitney class of $E_{\mathbb R}$ is the reduction mod $2$ of the total Chern class of $E_{\mathbb C}$. This is Problem 14-B on page 171 of Characteristic Classes, by John Milnor and James Stasheff $\endgroup$ – Alex Suciu Jan 21 '12 at 18:44

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

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