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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.

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    $\begingroup$ 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, 2016 at 14:19
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    $\begingroup$ 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, 2016 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, 2016 at 15:34

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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.

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  • $\begingroup$ 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, 2016 at 15:36

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