I wonder about the notion of a spin structure for varieties over any field and results in this direction. For example, I wonder if there is something like a spin-bundle for the sphere $x^2+y^2+z^2=R^2$ and the projective space qith $q^2+q+1$ elements for the finite fields $\mathbb{F}_q$ depending on $q$

Thanx in advance! Simon

PS: I have read something in Spin structures on schemes, and it seems the question has to do with an existence of a nondegenerate quadratic form on the

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    $\begingroup$ For curves, one could mimic Spin structures as square roots of $K$. $\endgroup$ – Alex Degtyarev Jun 20 '16 at 20:43
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    $\begingroup$ @BenMcKay, if not, then he may need a spin doctor. $\endgroup$ – LSpice Jun 20 '16 at 22:56

Let me work with arbitrary affine schemes $X = \text{Spec } k$. I believe there is a reasonable notion of a spin structure on a quadratic module $(V, q)$ over $k$, by which I mean a pair consisting of a finitely generated projective $k$-module (vector bundle over $X$) and a map $q : V \to k$ such that

  1. $q(\lambda v) = \lambda^2 q(v)$ for all $v \in V, \lambda \in k$, and
  2. $b(v, w) = q(v + w) - q(v) - q(w)$ is bilinear.

Namely, any such thing has an associated Clifford algebra $\text{Cl}(V, q)$, and a reasonable notion of spin structure in this context is that a spin structure on $(V, q)$ is a super Morita trivialization of $\text{Cl}(V, q)$, or somewhat more explicitly a finitely generated projective super $k$-module $M$ such that $\text{End}_k(M) \cong \text{Cl}(V, q)$ (here End is the super End) and such that $\text{Hom}_k(M, -)$ is faithful (or something like that).

In differential geometry the miracle is that any smooth manifold $M$ has a more-or-less canonically associated quadratic module (over $C^{\infty}(M)$), namely the tangent bundle of $M$ equipped with some Riemannian metric. The sense in which this is more-or-less canonical is that the space of Riemannian metrics on $M$ is contractible. But I don't see any analogue of this construction in algebraic geometry.

Edit: Following Alex Degtyarev's comment, here is a different candidate. One way to think about spin structures on a smooth manifold $M$ is that they are orientations with respect to real K-theory. Similarly, spin$^{\mathbb{C}}$ structures are orientations with respect to complex K-theory. The point of these orientations is that they provide pushforward maps in K-theory to a point, explicitly implemented by taking the index of a suitable Dirac operator.

But in algebraic geometry these pushforwards already exist: if $X$ is proper over a base $S$, then (maybe together with some other finiteness conditions) we can take the pushforward from coherent sheaves on $X$ to coherent sheaves on $S$. In particular, if $S = \text{Spec } \mathbb{C}$, pushforward takes vector bundles on a projective variety $X$ to finite-dimensional complex vector spaces. The relationship to the topological story is that smooth projective varieties over $\mathbb{C}$, or more generally complex manifolds, have a canonical spin$^{\mathbb{C}}$ structure.

Attached to a spin$^{\mathbb{C}}$ structure is a certain complex line bundle, and a choice of lift to a spin structure is precisely a choice of square root of this bundle. In the case of smooth projective varieties over $\mathbb{C}$ this line bundle is the canonical bundle, so one can think of spin structures on such things (compatible with their canonical spin$^{\mathbb{C}}$ structures) as square roots of canonical bundles (see e.g. the nLab for some details and references).

This suggests the following definition for a proper scheme $X$ over a base $S$: writing $f : X \to S$ for the anchor map, and $\omega_f \cong f^{!}(\mathcal{O}_S)$ for the dualizing complex, we might define a spin structure on $X$ to be a choice of square root of $\omega_f$. I don't know whether this definition is good for anything, but unlike the previous definition it applies to schemes and not just quadratic modules, and it recovers the usual notion of spin structure when applied to smooth projective varieties over $\mathbb{C}$.

  • $\begingroup$ Unless I'm missing something, what you're describing is more like a spin$^\mathbb{C}$ structure, no? Following Plymen, to get something like a spin structure, you'd presumably need $M$ isomorphic to $\operatorname{Hom}_k(M,k)$ as $\operatorname{Cl}(V,q)$-modules, where $\operatorname{Hom}_k(M,k)$ is given the $\operatorname{Cl}(V,q)$-module structure induced by the order-reversing anti-automorphism on $\operatorname{Cl}(V,q)$. $\endgroup$ – Branimir Ćaćić Jun 20 '16 at 21:47
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    $\begingroup$ @Branimir: I guess it depends on whether you have in mind the case that $k$ is a $\mathbb{C}$-algebra or an $\mathbb{R}$-algebra. $\endgroup$ – Qiaochu Yuan Jun 20 '16 at 21:55
  • $\begingroup$ Ah, OK, that makes sense. And even in the $\mathbb{C}$-algebra case, given that $\mathbb{CP}^{2k}$ is spin$^\mathbb{C}$ but not spin, I suppose you'd might as well work with something more precisely akin to spin$^\mathbb{C}$? $\endgroup$ – Branimir Ćaćić Jun 20 '16 at 22:03

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