Let ${C}$ be an algebraic curve over ${\mathbb{C}}$. A theta characteristic on ${C}$ is a (holomorphic or algebraic) square root of the canonical line bundle ${K_C}$, i.e. a line bundle ${L \in \mathrm{Pic}(C)}$ such that

$\displaystyle L^{\otimes 2} \simeq K_C.$

Since the degree of ${K_C}$ is even, such theta characteristics exist, and in fact form a torsor over the 2-torsion in the Jacobian ${J(C) = \mathrm{Pic}^0(C)}$, which is isomorphic to ${H^1(C; \mathbb{Z}/2\mathbb{Z}) \simeq (\mathbb{Z}/2\mathbb{Z})^{2g}}$.

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 ${M}$. On a framed manifolds, there is a canonical choice of quadratic refinement on the middle-dimensional mod ${2}$ 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 ${2}$ function ${L \mapsto \dim H^0(C, L)}$ on the theta characteristics is precisely this invariant. In other words, theta characteristics give a purely algebraic (valid in all characteristics, at least ${\neq 2}$) 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.

1. Examples

In genus two, every curve ${C}$ is hyperelliptic via the canonical map

$\displaystyle C \rightarrow \mathbb{P}^1,$

which is ramified at six points ${p_1, \dots, p_6 \in C}$. The canonical divisor has the property that

$\displaystyle K \sim 2 (p_i), \quad 1 \leq i \leq 6,$

so that the line bundles ${\mathcal{O}(p_i)}$ (which are pairwise linearly inequivalent) give six theta characteristics.

Since the theta characteristics form a torsor over the 2-torsion in the Jacobian, which is isomorphic to ${(\mathbb{Z}/2\mathbb{Z})^{4}}$, we should expect ten more theta characteristics. These will not be effective; for distinct ${i, j, k}$ with ${j < k}$, the line bundle corresponding to the divisor

$\displaystyle p_i + p_j - p_k,$

is a theta characteristic. (In fact, ${p_j - p_k}$ is a 2-torsion point in the Jacobian, and as ${j < k}$, these range over all the 15 nonzero 2-torsion points.) These form a (redundant) list of all the theta characteristics on ${C}$.

In genus three, given a theta characteristic ${L}$, we observe that ${L}$ has degree two, so ${H^0( L)}$ has dimension either ${0, 1, 2}$, and the last one occurs only if ${C}$ is hyperelliptic. So suppose ${C}$ is a nonhyperelliptic genus three curve, which means that the canonical map

$\displaystyle C \rightarrow \mathbb{P}( H^0( K_C)),$

imbeds ${C}$ as a smooth quartic in ${\mathbb{P}^2}$. In this case, there are the effective theta characteristics, each of which necessarily corresponds to a unique effective divisor ${p + q}$. To say that ${p + q}$ is a theta characteristic is to say that

$\displaystyle 2( p + q) \simeq K \simeq \mathcal{O}(1),$

under the canonical imbedding: that is, the intersection of ${C}$ with a line in ${\mathbb{P}^2}$ must cut out the subscheme ${2p + 2q \subset C}$. This means that the line is necessarily tangent to ${C}$ at both ${p,q}$, or in other words:

Proposition 1 Effective theta characteristics on the nonhyperelliptic genus three curve ${C}$ are in bijection with bitangent lines on ${C}$.

In fact, counting theta characteristics can be used to prove a fact from enumerative geometry, that a smooth plane quartic has exactly ${28}$ bitangents.

2. Spin structures and theta

The purpose of this section is to describe the following interpretation of theta characteristics in geometry:

Theorem 2 (Atiyah) On a compact complex manifold ${M}$, spin structures are in natural bijection with holomorphic square roots of the canonical bundle.

Proof: The holomorphic tangent bundle ${TM}$ is a complex vector bundle whose underlying ${\mathbb{R}}$-bundle is isomorphic to the usual real tangent bundle of ${M}$. In particular, it is a ${U(n)}$-bundle, and a spin structure consists of a lift of the underlying ${SO(n)}$-bundle, under the map

$\displaystyle U(n) \rightarrow SO(2n),$

to a ${\mathrm{Spin}(2n)}$-bundle under the double covering map ${\mathrm{Spin}(2n) \rightarrow SO(2n)}$; equivalently, it is a lift in the diagram

The choice of lifts to ${B \mathrm{Spin}(2n)}$ together with a homotopy to make the diagram commute) is canonically a ${H^1( M; \mathbb{Z}/2)}$-torsor. Since ${\mathrm{Spin}(2n)}$ pulls back to the unique two-fold cover ${\widetilde{U}(n)}$ of ${U(n)}$, to give a spin structure on ${M}$ is equivalent to giving the tangent bundle a reduction of structure group from ${U(n)}$ to ${\widetilde{U}(n)}$.

But since the determinant map

$\displaystyle U(n) \rightarrow U(1),$

induces an isomorphism on ${\pi_1}$, to give such a reduction of the structure group is equivalent to giving a reduction of structure group of the canonical bundle ${K_M}$ of top-forms under the double cover ${S^1 \rightarrow S^1}$. In other words, it is equivalent to giving a topological line bundle ${L}$together with a choice of isomorphism of topological bundles,

$\displaystyle L^{\otimes 2} \simeq K_M.$

But a choice of isomorphism determines a holomorphic structure on ${L}$, so that the squaring map to the total space of ${K_M}$ is holomorphic. In other words, it is equivalent to considering holomorphic bundles ${L}$ with a choice of isomorphism of holomorphic bundles

$\displaystyle L^{\otimes 2} \simeq K_M.$

However, since ${M}$ is compact, the “choice” of an isomorphism between holomorphic bundles is not really a choice: there is (if there is a choice) only a ${\mathbb{C}^* = \mathbb{G}_m}$‘s worth of choices. So there is not much extra data in choosing the isomorphism of holomorphic bundles ${L^{\otimes 2} \simeq K_M}$ (i.e., every complex number has a square root), and it’s equivalent to specifying ${L}$ with the holomorphic structure and not the map. $\Box$

3. Stability

The previous section showed that there was a purely algebro-geometric way of talking about “framings” on an algebraic curve ${C}$ over ${\mathbb{C}}$: they were in natural bijection with theta-characteristics on ${C}$. The second framed cobordism group (i.e., the second stable homotopy group ${\pi_2^s(S^0)}$) has a natural map

$\displaystyle \Omega^{\mathrm{fr}}_2 \stackrel{\simeq}{\rightarrow} \mathbb{Z}/2 ,$

given by the Kervaire invariant. Since framings correspond to theta characteristics, we should have an algebro-geometric way of obtaining an element of ${\mathbb{Z}/2}$ from a pair ${(C, L)}$ where ${L}$ is a theta characteristic.

Given a theta characteristic ${L}$ on ${C}$, one has the natural mod ${2}$ invariant

$\displaystyle (C, L) \mapsto \dim H^0(L) \mod 2,$

which turns out to be precisely the Kervaire invariant. In order to expect something like this, we’d have to show that the invariant ${(C, L) \mapsto \dim H^0(L) \mod 2}$ has good formal properties. For instance, the Kervaire invariant is constant in a family of framed manifolds, since the framed cobordism class in a smooth family does not vary.

In other words, we should expect the following:

Theorem 3 Given a family of smooth curves ${X \rightarrow B}$ and a line bundle ${\mathcal{L}}$ on ${X}$ such that ${\mathcal{L}^{\otimes 2}|_{X_b} \simeq K_{X_b}}$ for each ${b \in B}$, the function

$\displaystyle b \mapsto \dim H^0( X_b \mathcal{L}|_{X_b})$

is constant mod ${2}$.

In other words, given a family of curves and a continuously varying family of theta characteristics on them, the mod ${2}$ invariant constructed above is constant in the family. Note that the condition that ${\mathcal{L}^{\otimes 2}|_{X_b} \simeq K_{X_b}}$ for each ${b \in B}$ is equivalent, Zariski locally on the reduced base ${B}$, to the seemingly more natural or stronger condition

$\displaystyle \mathcal{L}^{\otimes 2} \simeq K_{X/B},$

as a fiberwise trivial line bundle on ${X}$ is the pull-back of a line bundle on ${B}$. This fact and related arguments are important in the theory of the relative Picard scheme of ${X \rightarrow B}$.

There seem to be (at least) two proofs of this. One argument, in Atiyah’s paper, relies on a mod 2 analog of the local constancy of the index of a Fredholm operator, by interpreting these ${H^0}$‘s as kernels of an appropriate ${\overline{\partial}}$-operator. There is also a purely algebraic proof of Mumford that reduces the result to a similar stability lemma for isotropic subspaces of a quadratic vector space.

After proving this, the analysis of theta characteristics on an arbitrary curve can be reduced to the analysis on a hyperelliptic curve, since the moduli space of curves is connected: for instance, one can count how many even and odd theta characteristics there are on any smooth curve by reducing to the (much simpler) hyperelliptic case.