






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


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.