June 23, 2013
Let be an algebraic curve over . A theta characteristic on is a (holomorphic or algebraic) square root of the canonical line bundle , i.e. a line bundle such that
Since the degree of is even, such theta characteristics exist, and in fact form a torsor over the 2-torsion in the Jacobian , which is isomorphic to .
One piece of geometric motivation for theta characteristics comes from the following observation: theta characteristics form an algebro-geometric approach to framings. By a theorem of Atiyah, holomorphic square roots of the canonical bundle on a compact complex manifold are equivalent to spin structures. In complex dimension one, a choice of a spin structure is equivalent to a framing of . On a framed manifolds, there is a canonical choice of quadratic refinement on the middle-dimensional mod homology (with its intersection pairing), which gives an important invariant of the framed manifold known as the Kervaire invariant. (See for instance this post on the paper of Kervaire that introduced it.)
It turns out that the mod function on the theta characteristics is precisely this invariant. In other words, theta characteristics give a purely algebraic (valid in all characteristics, at least ) approach to the Kervaire invariant, for surfaces!
Most of the material in this post is from two papers: Atiyah’s Riemann surfaces and spin structures and Mumford’s Theta characteristics of an algebraic curve. (more…)
October 4, 2012
This is the second post in a series on Kervaire’s paper “A manifold which does not admit any differentiable structure.” In the previous post, we described a form on the middle cohomology of a -connected -dimensional manifold, for . In this post, we can define the Kervaire invariant of such a framed manifold, by showing that this defines a form. I’ll try to sketch the proof that there is no framed manifold of Kervaire invariant one in dimension 10.
1. The form is a quadratic refinement
Let’s next check that the form defined in the previous post (we’ll review the definition here) is actually a quadratic refinement of the cup product. Precisely, this means that for , we want
In particular, this implies that descends to a function on , as it shows that of an even class is zero in . The associated quotient map is, strictly speaking, the quadratic refinement.
In order to do this, let’s fix . As we saw last time, these can be obtained from maps
by pulling back the generator in degree . Let be a map associated to , and let be a map associated to . We then have that
for the generator of . As we saw, this was equivalent to the definition given last time. (more…)