# Spin structures on schemes

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?

Thanks.

• 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 – Denis Nardin Apr 13 '16 at 14:19
• 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. – Donu Arapura Apr 13 '16 at 15:32
• @DonuArapura: I am precisely interested in the situation on schemes, not on manifold or complex manifolds, which is well-known. – Bilateral Apr 13 '16 at 15:34