This is a very naive question, but I have been wondering about the role of spin geometry and spinor structures in the context of algebraic geometry. I know the definition of spin structures and associated spinor bundles on manifolds and CW-complexes. The question is: is there analog notions of the following classical objects:

  • Spin structure

  • Bundle of Clifford algebras

  • Spinor bundle/bundle of Clifford modules

in the context of schemes (over the complex numbers) and which can be developed in a purely algebraic language? For example, is there a notion of "spin structure" or sheaf of Clifford algebras on a scheme?


  • 4
    $\begingroup$ I think that part of the problem is that defining a Spin structure requires a nondegenerate quadratic form on your bundle. In topology you can always choose a positive definite one (they're all equivalent anyway) but over a scheme the question is much more subtle $\endgroup$ – Denis Nardin Apr 13 '16 at 14:19
  • 2
    $\begingroup$ If I remember correctly (it's been a while) the usual condition for existence is of a spin structure on a manifold is for $w_2=0$. So for a complex manifold, it would be enough to know that $c_1$ is even, or that the canoncal bundle has a square root. These are classically called theta characteristics. This is probably different from what you are after, but I thought I'd mention it. $\endgroup$ – Donu Arapura Apr 13 '16 at 15:32
  • $\begingroup$ @DonuArapura: I am precisely interested in the situation on schemes, not on manifold or complex manifolds, which is well-known. $\endgroup$ – Bilateral Apr 13 '16 at 15:34

You might consider holomorphic spin structures on complex algebraic manifolds. If you try to reduce the frame bundle of a complex manifold to a holomorphic spin structure, you will induce a reduction to a holomorphic Riemannian metric, giving a holomorphic affine connection, forcing the vanishing of the Atiyah class, and therefore the vanishing of all Chern classes. Therefore the manifold admits a finite unramified covering by a complex torus, and the holomorphic Riemannian metric is translation invariant. So you don't want a holomorphic spin structure, unless you want to allow singularities.

  • $\begingroup$ I am precisely interested in the case where there are singularities. In particular I would like to know if there is a reasonable intrinsic definition of spin structure or spinor bundle on a scheme. Intrinsic in the sense that it can be defined in algebraic terms and without any reference to any underlying smooth manifold, which may not exist. $\endgroup$ – Bilateral Apr 13 '16 at 15:36

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.