TITLE: Spin structures on schemes
QUESTION [8 upvotes]: 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.

REPLY [4 votes]: 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.