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Jacobi’s inversion theorem

May 11, 2014

In this post, I will discuss Jacobi’s inversion theorem. It is a follow up in a series of posts about Riemann surfaces, the first of which being this one. This theorem tells us in what sense the Abel-Jacobi map \mu is surjective.

Theorem (Jacobi’s inversion): Let S be a compact Riemann surface of genus g and take any p_0 \in S. Then for any \lambda \in J(S), we can find g points p_1, \dots, p_g \in S such that

\mu\left(\displaystyle\sum_{i=1}^g(p_i-p_0)\right) = \lambda.

In other words, for \omega_1, \ldots, \omega_g a basis of H^0(S,\Omega^1), for all \lambda \in \mathbb{C}^g, there is p_1, \ldots, p_g \in S and paths \alpha_i from p_0 to p_i such that

\displaystyle\sum_{i=1}^g \int_{\alpha_i}\omega_j = \lambda_j     for all 1\leq j \leq g.

With this theorem coupled with Abel’s theorem, we will have completely proved the exactness of

\mathcal{M}(S) \overset{()}\to \text{Div}_0(S) \overset{\mu}\to J(S) \to 0.

Lemma 1: The set S^{(d)} of effective divisors of degree d on S is a compact complex manifold.

Proof of lemma 1: Consider the action of the permutation group \mathfrak{S}_d on

S^d = S \times \cdots \times S    (d times).

The quotient, denoted S^d/\mathfrak{S}_d = \text{Sym}^d(S), inherits the quotient topology making \pi : S^d \to \text{Sym}^d(S) a continuous map. Clearly, \text{Sym}^d(S) is in bijection with S^{(d)}. Suppose for a moment that S = \mathbb{C}. We will write an element of \text{Sym}^d(\mathbb{C}) as a sum \lambda_1 + \cdots \lambda_d to indicate the unimportance of the ordering (note that the \lambda_i‘s are not necessarily distinct). We consider the map

\varphi : \text{Sym}^d(S) \to \mathbb{C}^d

that takes \sum_{i=1}^d\lambda_i to the d-tuple (a_1, \ldots, a_d) \in \mathbb{C} consisting of the coefficents of the monic polynomial having \lambda_1, \ldots, \lambda_d as its roots, i.e

\varphi(\lambda_1 + \cdots \lambda_d) = (a_1, \ldots, a_d)

if

z^d + a_1z^{d-1} + \cdots a_{d-1}z + a_d = (z-\lambda_1)\cdots (z-\lambda_d).

In other words, \varphi(\{\lambda_i\}) = (\sigma_1(\{\lambda_i\}), \ldots, \sigma_d(\{\lambda_i\}) where \sigma_i is the ith elementary symmetric polynomial. By the fundamental theorem of algbera, \varphi : \text{Sym}^d(\mathbb{C}) \to \mathbb{C}^d is a bijection, and in fact a homeomorphism.

The difficulty is in seeing that \varphi^{-1} is also continuous. To see this, take an open set \pi (U) \subset \text{Sym}^d(\mathbb{C}) and consider (a_1, \ldots, a_d) \in V := \varphi (\pi (U)) \subset \mathbb{C}^d. In other words,

p_0 := z^d + a_1z^{d-1} + \cdots + a_{d-1} + a_d = \displaystyle\prod_{i=1}^d(z-\lambda_i)     with \lambda_i \in U.

We need to show that there is an open set W \subset V. Write

N_U(p) = \dfrac{1}{2\pi\sqrt{-1}}\displaystyle\int_{\partial U} \frac{dp}{p}

which, when defined, is the number of zeros of the polynomial p inside U. We can choose an open set W \subset \mathbb{C}^d containing p_0 small enough so that N_U(p) is defined for all p \in W. Then since p \mapsto N_U(p) is continuous and takes only integer values, we have

N_U(p) = N_U(p_0) = d    for all p \in W.

But this means that for all p \in W, all the roots of p are in U so that p_0 \subset W \subset V as we wanted.

This gives \text{Sym}^d(\mathbb{C}) the structure of a complex manifold of dimension d, in fact biholomorphic to \mathbb{C}^d.

Now we put a similar complex manifold structure on \text{Sym}^d(S) for arbitrary S. Consider D = \sum_{i=1}^dp_i \in \text{Sym}^d(S) and take holomorphic charts (U_i,z_i) around p_i on S such that U_i \cap U_j = \emptyset if p_i \neq p_j and U_i = U_j, z_i = z_j if p_i = p_j. We obviously have an injective mapping

\varphi_D : \pi(U_1 \times \cdots U_d) \to V_D \subset \mathbb{C}^d

where \pi : S^d \to \text{Sym}^d(S), by doing as above;

\varphi_D(\sum_jz_j) = \left(\sigma_1(\{z_j\}), \ldots, \sigma_d(\{z_j\})\right).

The verifications that this gives a homeomorphism onto an open set of \mathbb{C}^d is just as in the earlier case. These maps thus provide \text{Sym}^d(S) with a holomorphic atlas.

QED

Fixing a base point p_0 \in S we get a (holomorphic) injection

\iota : S^{(d)} \hookrightarrow \text{Div}_0(S)

\iota : \displaystyle\sum_{i=1}^d p_i \mapsto \sum_{i=1}^d(p_i-p_0)

and thus holomorphic mappings

\mu^{(d)} : S^{(d)} \to J(S)

\mu^{(d)} : \displaystyle\sum_{i=1}^dp_i \mapsto \sum_{i=1}^d \left(\int_{p_0}^{p_i}\omega_1, \ldots, \int_{p_0}^{p_i}\omega_g\right)    (mod \Lambda)

by composing \mu and \iota. Jacobi’s inversion theorem says that \mu^{(g)} is surjective.

Lemma 2:  Let f : M \to N be a holomorphic map between two compact connected complex manifolds of the same dimension. If f is not everywhere singular, i.e. the Jacobian matrix J(f) is not identically zero, then f is necessarily surjective.

Proof of lemma 2: This is immediate from the proper mapping theorem which says that if V \subset M is an analytic subvariety, then f(V) is an analytic subvariety of N. In this case, f(M) would be a compact subvariety containing an open set which would mean f(N) = M. Griffiths-Harris presents a more elementary proof which does not use the rather deep proper mapping theorem:

Consider \psi_N a volume form on N. Since J(f) is not identically 0 and since f preserves the orientation (being holomorphic), we have

\displaystyle\int_Mf^*\psi_N > 0.

Since for any q \in N we have H^{2n}(N-\{q\},\mathbb{R}) = 0, the volume form is exact in N - \{q\} and

\psi_N = d\varphi

for some (2n-1)-form \varphi on N - \{q\}. But then if q \notin f(M), we have

\displaystyle\int_M f^*\psi_N = \int_{\partial M}df^*\varphi = 0,

which contradicts earlier considerations.

QED

To prove the theorem, we thus have to show that \mu^{(g)} is not everywhere singular.

Proof of the theorem: At points D = \sum_ip_i \in S^{(d)} such that the p_i‘s are distinct, the quotient map \pi : S^d \to S^{(d)} is locally a biholomorphism. So choosing disjoint charts (U_i,z_i) in S centered at p_i, we get a chart

\varphi : \pi(U_1 \times \cdots \times U_d) \to \mathbb{C}^d

\varphi(\sum_iq_i) = (z_1(q_1), \ldots, z_d(q_d)).

In such coordinates, for D' = \sum_iq_i near D, we have

\mu^{(g)}(D') = \displaystyle\sum_{i=1}^g\left(\int_{p_0}^{z_i}\omega_1, \ldots, \int_{p_0}^{z_i}\omega_g \right)     mod \Lambda.

So

\dfrac{\partial}{\partial z^i}(\mu^{g}_j(D')) = \displaystyle\sum_{k=1}^g \dfrac{\partial}{\partial z^i}\left(\int_{p_0}^{z_k}\omega_j\right).

But with \omega_j(z^i) = h_{ji}(z^i)dz^i near p_i, this is

\dfrac{\partial}{\partial z^i} \displaystyle\int_{p_0}^{z_i}h_{ji}(z^i)dz^i

hence

\dfrac{\partial}{\partial z^i} (\mu^{(g)}_j(D')) = h_{ji}(z^i).

The jacobian matrix of \mu^{(d)} near D is thus given by

J(\mu^{(d)}) = \left(\begin{array}{ccc}h_{11} & \cdots & h_{1d}\\ \vdots && \vdots \\ h_{g1} & \cdots & h_{gd}\end{array}\right)_{g\times d}.

It suffices to see that this matrix is of full rank for some choice of D = \sum_{i=1}^d p_i and basis \omega_1, \ldots, \omega_g. We simply do Gauss reduction: Choose p_1 such that \omega_1(p_1) \neq 0. Then subtracting a multiple of \omega_1(p_1) to \omega_2, \ldots, \omega_g, we make it so that

\omega_2(p_1) = \cdots = \omega_g(p_1) = 0.

The \omega_i‘s are still a basis of \Omega^1(S) and we continue like this, choosing a p_2 such that \omega_2(p_2) \neq 0 etc… We eventually arrive at the form

J(\mu^{(d)}) = \left(\begin{array}{cccccc}h_{11}&h_{12}&\cdots&h_{1g}&\cdots&h_{1d} \\ 0&h_{22}&\cdots&h_{2g}&\cdots&h_{2d} \\ \vdots&\vdots&\ddots&&& \\ 0&0&\cdots&h_{gg}&\cdots&h_{gd}\end{array}\right)

which is of maximal rank since h_ii(p_i) \neq 0 for all 1 \leq i \leq g. This shows that \mu^{(g)} : S^{(g)} \to J(S) is not everywhere singular, so it is surjective by lemma 2.

QED

Putting everything together, we showed that the set of (isomorphism classes) of topologically trivial holomorphic line bundles L \to S (i.e. with zero first Chern class) has the structure of a complex torus \mathbb{C}^g/\Lambda of dimension g. Indeed, the group \text{Pic}_0(S) consisting of those (isomorphism classes of) holomorphic line bundles is isomorphic to \text{Div}_0/\mathcal{M}(S), which by Abel’s and Jacobi’s theorem is isomorphic to J(S).

Note that in fact the fibres of \mu^{(g)} : S^{(g)} \to J(S) consist of projective spaces. Indeed, if \mu^{(g)}(D) = \lambda, then by Abel’s theorem the fiber is (\mu^{(g)})^{-1}(\mu(D)) = |D|, the set of effective divisors linearly equivalent to D, which corresponds to the projectivisation of H^0(S,\mathcal{O}(D)). It can be shown that generically the fiber of \mu^{(g)} is a point and that \mu^{(g)} is a birational map.

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