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Abel’s Theorem

May 9, 2014

In the first post of this series, I defined the Abel-Jacobi map \mu : \text{Div}_0(S) \to J(S) from a compact Riemann surface S of genus g \geq 1 to its jacobian J(S). Recall that the jacobian of S is defined to be the complex torus \mathbb{C}^g/\Lambda where \Lambda is the period lattice

\Lambda = \left\{\displaystyle\sum_{i=1}^{2g}\alpha_i\Pi_i \mid \alpha_i \in \mathbb{Z}\right\}

for \Pi_i the period vectors

\Pi_i = \left( \displaystyle\int_{\delta_i}\omega_1, \ldots, \int_{\delta_i}\omega_g\right) \in \mathbb{C}^g

associated to some basis \omega_1, \ldots, \omega_g of the space \Omega^1(S) of holomorphic 1-forms, and where \delta_1, \ldots, \delta_{2g} are cycles giving a canonical basis of H_1(S,\mathbb{Z}). For D = \sum_{i=1}^d(p_i-q_i) a divisor of degree 0, the Abel-Jacobi map is then

\mu(D) = \displaystyle\sum_{i=1}^d\left(\int_{q_i}^{p_i}\omega_1, \ldots, \int_{q_i}^{p_i}\omega_g\right)    (mod \Lambda).

The goal of this post of to prove the following result:

Theorem (Abel): Let D \in \text{Div}_0(S) and suppose \omega_1, \dots, \omega_g is a basis of \Omega^1(S). Denote by \mathcal{M}(S) the meromorphic functions on S. Then D = (f) for some f \in \mathcal{M}(S) if and only if \mu(D) = 0 \in J(S).

Proof: The only if part is quite easy. Suppose D is the divisor associated to a meromorphic function f \in \mathcal{M}(S), i.e. D = (f) = (f)_0 - (f)_{\infty} for (f)_0, (f)_{\infty} respectively the zeros and poles of f. The idea is to view f as a holomorphic function

f : S \to \mathbb{CP}^1.

Then f has a well-defined degree d which is the number of points (counted with multiplicity) in a fiber f^{-1}(p). Let’s note D(t) = f^{-1}(t) the divisor which is the fiber of f at t \in \mathbb{CP}^1. Then the degree of (D(t) - dp_0) is zero for all p_0 \in S, and

\mu(D(t) - dp_0) = \left(\displaystyle\int_{\sigma}\omega_1, \ldots, \int_{\sigma}\omega_g\right)    (mod \Lambda)

for \sigma some chain in S with \partial \sigma = D(t) - dp_0. Since the set of points in f^{-1}(t) vary analytically with t (f is a locally z^k with 1 \leq k \leq d), we obtain a holomorphic function

\varphi : \mathbb{CP}^1 \to J(S)

by defining

\delta(t) = \mu(D(t) - dp_0).

But since \mathbb{CP}^1 is simply connected, it lifts to a holomorphic function \tilde{\varphi} : \mathbb{CP}^1 \to \mathbb{C}^g, which must be constant by the maximum principle. Hence \varphi itself is constant, which means in particular that \varphi(0) = \varphi(\infty), i.e. that

\mu(D(0)) - \mu(D(\infty)) = 0.

Since \mu is additive and D(0) - D(\infty) = (f)_0 - (f)_{\infty} = (f), this tells us that \mu(D) = 0 when D = (f), which is what we wanted to show.

The converse is harder. Given a divisor D = \sum_i(a_ip_i-b_iq_i) of degree 0, where a_i, b_i \in \mathbb{Z}_{>0} and the p_i,q_i are all distinct, such that \mu(D) = 0, we search for a meromorphic function f \in \mathcal{M}(S) such that (f) = D. We first reduce the problem to the existence of a certain differential of the third kind:

Lemma 1: There is an f \in \mathcal{M}(S) with (f) = D if and only if there is a meromorphic 1-form \eta \in (\Omega^1 \otimes \mathcal{M})(S) such that

1. (\eta)_{\infty} = -\left(\displaystyle\sum_i(p_i+q_i)\right),

i.e. \eta has a simple pole exactly at the zeros and poles of f;

2. \text{res}_{p_i}(\eta) = \dfrac{a_i}{2\pi\sqrt{-1}}      and      \text{res}_{q_i}(\eta) = \dfrac{-b_i}{2\pi\sqrt{-1}},

i.e. the residues at those poles are given by the order of f;

3. \int_{\gamma}\eta \in \mathbb{Z} for any loop \gamma in S \backslash \{p_i, q_i\}.

The correspondence being given by

f \mapsto \eta = \dfrac{1}{2\pi\sqrt{-1}}\dfrac{df}{f}

and

\eta \mapsto f(p) = \exp\left(2\pi\sqrt{-1}\displaystyle\int_{p_0}^p\eta\right)

for any p_0 \in S.

Proof of lemma 1: Let f be such a meromorphic function and define \eta as above. Then since f is locally given by f(z) = z^k for some k \in \mathbb{Z}, we have df/f = k/z locally, with k = \text{ord}(f). So df/f has no zero and a simple pole at every zero and pole of f, with residue at a point equal to the order of f at that point. Conditions 1. and 2. are thus satisfied for this \eta. To see that the periods of \eta are integers, we write

[\gamma] = \displaystyle\sum_iw_i[C_{p_i}] + w_i'[C_{q_i}]

for a loop \gamma in S \backslash \{p_i,q_i\} where C_{p_i},C_{q_i} are small loops around p_i and q_i. Then

\displaystyle\int_{\gamma}\eta = \dfrac{1}{2\pi\sqrt{-1}}\displaystyle\sum_i\left(w_i\int_{C_{p_i}}d\log f + w_i' \int_{C_{q_i}}d\log f\right) = \sum_i w_i + w_i' \in \mathbb{Z}.

Conversely, given a meromorphic 1-form \eta satisfying conditions 1., 2. and 3., we let f be the function

f(p) = \exp\left(2\pi\sqrt{-1}\displaystyle\int_{p_0}^p\eta\right)

for some p_0 \in S not in \{p_i,q_i\}. Note that condition 3. insures that this function is well defined. Near one of \eta‘s simple pole p_i, we can write

\eta = \dfrac{a_i}{2\pi\sqrt{-1}}\dfrac{dz}{z} + h(z)dz

for h(z) some never-vanishing holomorphic function around p_i. Then for p_0^i sufficiently near p_i, we have in these coordinates

f(z) = \exp\left(2\pi\sqrt{-1}\left[\displaystyle\int_{p_0}^{p_0^i}\eta + \int_{p_0^i}^z\frac{a_i}{2\pi\sqrt{-1}}\frac{dz}{z} + \int_{p_0}^zh(z)dz\right]\right)

which gives

f(z) = H_1(z)\exp\left(a_i\displaystyle\int_{p_0}^zd\log (z)\right) = H_2(z)\exp\left(a_i\log (z)\right) = z^{a_i}H(z)

for H_1, H_2, H never vanishing holomorphic functions around p_i. Similarly around a point q_i we find f(z) = z^{-b_i}H(z) and we conclude that

(f) = \displaystyle\sum_i(a_ip_i - b_iq_i),

concluding the proof of the lemma

QED

To construct such a meromorphic 1-form, we will use another lemma:

Lemma 2: Given a finite number of points \{p_i\} on S and complex numbers a_i \in \mathbb{C} such that \sum_ia_i = 0, there exists a meromorphic 1-form \eta with only simple poles having its poles exactly at the points p_i with residue a_i at p_i.

Proof of lemma 2: We consider the exact sequence of sheaves

0 \longrightarrow \Omega^1 \longrightarrow \Omega^1(\sum_ip_i) \overset{\text{res}}\longrightarrow \bigoplus_i\mathbb{C}_{p_i} \longrightarrow 0

where \Omega^1(\sum_ip_i) is the sheaf of meromorphic 1-forms having only simple poles exactly at the points p_i, i.e. if D = \sum_ip_i, then \Omega^1(\sum_ip_i) = \Omega^1 \otimes_{\mathcal{O}_S} \mathcal{O}([D]), and where \mathbb{C}_{p_i} is the skyscraper sheaf around p_i. By Kodaira-Serre duality (see for example p. 153 in Griffiths & Harris, replacing E with the trivial line bundle),

H^1(S,\Omega^1) \cong H^0(S,\mathcal{O}) \cong \mathbb{C}

where the last isomorphism comes from the maximum principle. Then the long exact sequence gives the exact sequence

H^0(S,\Omega^1(\sum_ip_i)) \overset{\text{res}}\longrightarrow H^0(S,\bigoplus_i\mathbb{C}_{p_i}) \longrightarrow H^1(S,\Omega^1)

i.e.

H^0(S,\Omega^1(\sum_ip_i)) \overset{\text{res}}\longrightarrow \bigoplus_i \mathbb{C} \to \mathbb{C}

so the residue map has 1-dimensional cokernel, i.e. the image of H^0(S,\Omega^1(\sum_ip_i)) is of codimension at most 1. But if \eta \in H^0(S,\Omega^1(\sum_ip_i)), then by the residue theorem \sum_i \text{res}_{p_i}(\eta) = 0 hence the image of H^0(S,\Omega^1(\sum_ip_i)) by the residue map, which is a linear subspace of codimension at most 1, is contained in the hyperplane \{\sum_ia_i = 0\} \subset \bigoplus_i\mathbb{C} which is of codimension 1. Hence these two sets are equal, i.e.

\text{res}\left(H^0(S,\Omega^1(\sum_ip_i))\right) = \{(a_i)_i \in \bigoplus_i\mathbb{C} \mid \sum_ia_i = 0\}

which is exactly what the lemma says.

QED

Back to the proof of the theorem: Given our divisor D = \sum_i(a_ip_i - b_iq_i) of degree 0, this last lemma tells us we can find a meromorphic 1-form \eta satisfying conditions 1. and 2. of the first lemma. All that remains to be done is to show that we can perturb this \eta such that its periods are integers, i.e. such that it satisfies condition 3. of the lemma. Since the biholomorphism class of J(S) is independant of the choice of basis \delta_1, \ldots, \delta_{2g} for H_1(S,\mathbb{Z}) and \omega_1, \ldots, \omega_g for \Omega^1(S), we may take (\delta_i) a canonical basis and (\omega_i) normalised with respect to this basis, i.e. such that

\displaystyle\int_{\delta_i}\omega_i = \delta_{ij}      for 1 \leq i,j \leq g.

Recall from last post that such a choise of basis for \Omega^1(S) is possible as a consequence of the reciprocity law. Let \eta satisfy conditions 1. and 2. and denote its periods by

N^i = \displaystyle\int_{\delta_i}\eta      (i=1, \ldots, 2g).

By correcting \eta with a linear combinations of the \omega_i‘s, we can suppose its A-periods vanish. Indeed, take

\eta' = \eta - \displaystyle\sum_{i=1}^g N^i \omega_i.

Then \eta' still satisfies conditions 1. and 2. because the \omega_i‘s are holomorphic, but clearly for i = 1, \ldots, g, we have \int_{\delta_i}\eta' = 0.

The game is now to add an integral linear combination of the \omega_i‘s to \eta to make its B-periods integers. By the reciprocity law,

\displaystyle\sum_{i=1}^g \left(N^{i+g}\int_{\delta_i}\omega_j - N^i\int_{\delta_{i+g}}\right) = 2\pi\sqrt{-1}\sum_{p \text{ pole of } \eta}\text{res}_{p}(\eta) \int_{p_0}^p\omega_j.

So since N^i = 0 for i \geq g and the basis (\omega_i) is normalised, by condition 2. we find

N^{j+g} = \displaystyle\sum_ia_i\int_{p_0}^{p_i}\omega_j - b_i\int_{p_0}^{q_i}\omega_j = \sum_i\int_{q_i}^{p_i}\omega_j    for all j = 1, \ldots, g

for a proper choice of path in this last integral (take  path from q_i to p_i circling the right amount of times along the \delta_j‘s to incorporate the a_i,b_i‘s in the integral). Let us denote by \gamma_i those paths on which we integrate in this last expression. Since

\mu(D) = \displaystyle\sum_i\left(\int_{\gamma_i}\omega_1, \ldots, \int_{\gamma_i}\omega_g\right) \in \Lambda \subset \mathbb{C}^g

by hypothesis, there exists a cycle

\sigma \sim \displaystyle\sum_{k=1}^{2g}m_k\delta_k        with m_k \in \mathbb{Z}

such that

\displaystyle\sum_i\int_{\gamma_i}\omega_j = \int_{\sigma}\omega_j      for all j=1, \ldots, g

(this is the definition of being 0 in the jacobian). Then we have

N^{g+j} = \displaystyle\sum_i\int_{\alpha_i}\omega_j = \int_\sigma\omega_j     for all j = 1, \ldots, g.

The periods of \eta are thus

N^i = 0

and

N^{g+i} = \displaystyle\sum_{k=1}^{2g}m_k\int_{\delta_k}\omega_i = m_i + \sum_{k=1}^gm_{g+k}\int_{\delta_{g+k}}\omega_i

for i=1, \ldots, g. We can now correct \eta for it to satisfy conditions 1. 2. and having integral B-periods by taking

\eta' = \eta - \displaystyle\sum_{k=1}^gm_{g+k}\omega_k.

Indeed, for i=1, \ldots, g, the periods of \eta' are

N'^i = -m_{g+1}

and

N'^{g+i} = N^{g+i} - \displaystyle\sum_{k=1}^gm_{g+k}\int_{\delta_{g+i}}\omega_k = m_i + \sum_{k=1}^gm_{g+k}\left(\int_{\delta_{g+k}}\omega_i - \int_{\delta_{g+i}}\omega_k\right).

But by Riemann’s first bilinear relation, the expression in parentheses above vanishes for all 1 \leq i,k \leq g, so N'^{g+i} = m_i. We have thus found a meromorphic 1-form \eta satisfying conditions 1, 2 and 3 of lemma 1, concluding the proof of Abel’s theorem

QED

Corollary: For a compact Riemann surface of genus g \geq 1, the Abel-Jacobi map gives a holomorphic embedding of S into J(S) (i.e. it is injective and has maximal rank 1 everywhere)

Proof: To see that \mu|_S is injective, suppose that \mu(p) = \mu(q) for two distinct points p,q \in S, i.e. that \mu(p-q) = 0 \in J(S). Then by Abel’s theorem, there is a meromorphic function f \in \mathcal{S} having divisor (f) = p-q. We see this meromorphic function a holomorphic function

f : S \to \mathbb{CP}^1.

Recall that any holomorphic function between two compact Riemann surfaces is in fact a branched cover. In particular, f has a degree d (the degree is the number of points in the fibers of f, which, counting with multiplicity, does not depend on the point in the image). Since f^{-1}(\infty) = \{q\}, we have \text{deg}(f) = 1. But this would mean that \#f^{-1}(p) = 1 for every p \in \text{Im}(f) = \mathbb{CP}^1. So f would be a biholomorphism, which constradicts the fact that the genus is not 0.

To see that \mu has maximal rank everywhere, we just compute its differential. Let z be a holomorphic coordinate centered at p \in S and write \mu = (\mu_1, \ldots, \mu_g). Writing the basis of holomorphic one-forms in this chart as \omega_j = \eta_jdz, we have

\mu_j(z) = \displaystyle\int_{p_0}^p\omega_j + \int_0^z\eta_j(z)dz

so

\dfrac{\partial \mu_j}{\partial z} = \eta_j(z).

Since \omega_1, \ldots, \omega_g is a basis for \Omega^1(S), they never all vanish simultaneously, so \mu|_S has maximal rank 1 everywhere.

QED

Corollary: Every smooth Riemann surface T of genus one is biholomorphic to a complex torus \mathbb{C}/\Lambda.

Proof: This is immediate from last corollary: if g=1, the Abel-Jacobi map gives a biholomorphism

\mu : T \to J(T) \simeq \mathbb{C}/\Lambda.

QED

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