Where we stand, and the question this guide answers
By now this rung has given you a compact Riemann surface X of genus g as one object with three faces. Guide 3 taught you to bookkeep zeros and poles with a divisor D — a finite formal integer combination of points — and to declare two divisors linearly equivalent when their difference is the divisor of a meromorphic function. Guide 3 also fixed the dimension of the space of holomorphic differentials at exactly g, and packaged the counting of functions with prescribed poles into the Riemann-Roch theorem. This final guide ties the threads: it turns 'which divisors are linearly equivalent' from a question you check case by case into a single map into a torus, and then opens the door to the space of all genus-g surfaces.
Keep your two running examples in hand throughout. On the sphere CP^1 (genus 0) there are no holomorphic differentials at all, so the construction below collapses to a point — a useful sanity floor. On a torus C/L (genus 1) there is exactly one holomorphic differential, namely dz, and the whole machinery reduces to something you can almost see with your eyes: the Jacobian of the torus is the torus itself. The genus-1 case is not a toy; it is the honest model from which the general picture is the natural generalization. Lead with it whenever the higher-genus statement feels abstract.
The period lattice and the Jacobian
Start with the g-dimensional complex vector space of holomorphic 1-forms, which we write H = H^0(X, Omega). Pick a basis omega_1, ..., omega_g of it. Now recall from algebraic topology that the first homology H_1(X) of a genus-g surface is a free abelian group of rank 2g, with the standard generators a_1, b_1, ..., a_g, b_g of the handles. The bridge between the analytic side (forms) and the topological side (loops) is integration: a closed loop gamma and a holomorphic form omega produce the complex number integral over gamma of omega, called a period. Integrating each basis form over each basis loop fills out a 2g-by-g array of periods, and the columns it spans inside C^g form a lattice L, the period lattice.
The decisive fact — and it is a theorem, not a definition — is that these 2g period vectors are linearly independent over the reals, so L is a genuine full-rank lattice in C^g (real dimension 2g). This rests on the Riemann bilinear relations, which come from applying Stokes' theorem to wedge products of forms on a cut-open surface; the antisymmetric relation forbids degeneracy and the positive-definite relation pins down the orientation. We will not prove them here — that is honestly a few pages of careful cut-and-paste topology plus the Stokes theorem you met in the forms track — but the upshot is clean and is what you must remember.
Quotient the vector space by the lattice and you obtain the Jacobian variety of X, the Jacobian J(X) = C^g / L. It is a compact complex torus of dimension g, and the Riemann relations upgrade it to an abelian variety — a complex torus that actually embeds in projective space, carrying a polarization. For the genus-1 torus C/L, the space of forms is one-dimensional (spanned by dz), the period lattice is L itself, and J(X) = C/L = X: the torus is its own Jacobian. That coincidence is special to genus 1; in general J(X) is a g-dimensional object built from the one-dimensional X, and the two are emphatically not equal.
The Abel-Jacobi map: giving each point an address
Now build the map that justifies all this. Fix a basepoint p_0 in X. To any other point p, assign the vector of integrals of the basis forms along a path from p_0 to p. The trouble is obvious: the answer depends on which path you take, and two paths differ by a loop, which shifts the value by a period. But a period is exactly an element of L — so the ambiguity disappears the moment you read the answer modulo L. That is the entire idea of the Abel-Jacobi map u: X -> J(X), and the quotient by the period lattice is doing precisely the job of killing the path-dependence.
Fix basepoint p_0. Basis of holomorphic 1-forms: omega_1, ..., omega_g.
u(p) = ( integral_{p_0}^{p} omega_1 , ... , integral_{p_0}^{p} omega_g ) mod L
Different path -> differs by a loop gamma in H_1(X)
shift = ( integral_gamma omega_1 , ... , integral_gamma omega_g ) in L
So u(p) is well defined in J(X) = C^g / L.
Extend additively to divisors: u( sum n_i p_i ) = sum n_i u(p_i).Extend u from points to divisors by additivity: u of a divisor sum n_i p_i is just sum n_i u(p_i). This linear extension is the workhorse, because the interesting question — when are two divisors linearly equivalent? — is a question about divisors, not single points. Restricted to degree-zero divisors (those whose coefficients add to zero), the extended map is independent of the chosen basepoint, which is why the cleanest statements live on the degree-zero part. Keep firmly in view that u is a holomorphic map, and for genus at least 1 it is an embedding of X into its Jacobian — the curve sits inside the torus as a g-dimensional gallery's single curved wall.
Abel's theorem and the Jacobi inversion problem
Here is the payoff, and it is one of the most beautiful theorems in the subject. Abel's theorem: a degree-zero divisor D is the divisor of a meromorphic function — that is, D is principal — if and only if u(D) = 0 in the Jacobian. Combined with the additivity, this says two divisors of the same degree are linearly equivalent exactly when they have the same image under the Abel-Jacobi map. The vague guide-3 relation 'linearly equivalent' is thereby converted into a sharp equation in a torus: D_1 ~ D_2 iff u(D_1) = u(D_2). All the case-by-case checking dissolves into reading off a single point of J(X).
Abel's theorem tells you the fibres of u — which divisors map to the same point. The complementary question is surjectivity: does every point of J(X) actually arise? This is the Jacobi inversion problem, and the answer is yes — the map sending an unordered g-tuple of points (an effective divisor of degree g) to its image in J(X) is surjective, and generically one-to-one. So the Jacobian is not merely a receptacle; it is exactly parametrized by degree-g divisor classes. Together, Abel (the fibres) and Jacobi inversion (the surjectivity) identify J(X) with the group of degree-zero divisor classes, the object an algebraic geometer calls Pic^0(X).
How do you actually invert — given a point of the Jacobian, find the divisor? The classical answer is the Riemann theta function, an explicit holomorphic function on C^g built from the period matrix, whose zero set (the theta divisor) is translated into position to read off the g points. We will be honest and not develop theta functions here; they are a guide of their own, and the formulas are intricate. The structural takeaway is enough for now: the Jacobian linearizes the curve. Nonlinear questions about meromorphic functions and divisors become linear (additive) questions in a torus, which is why the Abel-Jacobi map is the organizing idea of the whole theory.
A glimpse of moduli: how all genus-g surfaces fit together
Step up one floor. We have been studying one surface X. Now ask: how many genus-g Riemann surfaces are there, and how do they vary? Genus is the discrete topological label, but two surfaces with the same genus can be inequivalent as complex manifolds — they carry different complex structures. The space whose points are isomorphism classes of compact Riemann surfaces of genus g is the moduli space M_g, the moduli space of curves. Counting its dimension is a famous Riemann calculation: M_0 is a single point (every genus-0 surface is CP^1), M_1 is one-complex-dimensional (the parameter tau of the torus C/(Z + Z*tau), up to the action of SL(2,Z)), and for g at least 2 the dimension is 3g - 3.
Where does 3g - 3 come from? One honest route is the Riemann-Roch theorem applied to the canonical class: the deformations of a complex structure are governed by a cohomology group whose dimension Riemann-Roch computes as 3g - 3 for g at least 2. Another, more geometric route uses the uniformization theorem from guide 4: a genus-at-least-2 surface is a hyperbolic surface, the quotient of the disk by a discrete group, and you can cut it into pairs of pants and reassemble — counting the gluing parameters (the Fenchel-Nielsen lengths and twists) gives 6g - 6 real, i.e. 3g - 3 complex. The two counts agreeing is a small miracle that ties the rung together: the analysis of guide 3 and the geometry of guide 4 see the same number.
There is a subtlety you must not paper over: M_g is not a manifold in the naive sense, because some surfaces have extra automorphisms (symmetries) and the moduli space has mild singularities at those points — it is an orbifold, or in algebraic language a stack. A cleaner upstairs object is Teichmuller space T_g, the Teichmuller space, which records a surface together with a marking (a choice of generators of pi_1 up to homotopy). T_g is a genuine complex manifold, diffeomorphic to a ball of complex dimension 3g - 3, and M_g is its quotient by the mapping class group, the group of marking-changes. So the orbifold messiness of M_g is exactly the price of forgetting the marking — a clean ball divided by a group with fixed points.
Honest cautions, and where the ladder goes from here
A few honesties to carry up. First, almost everything here needs compactness and genus at least the stated bound — the Jacobian construction wants a compact surface, the dimension 3g - 3 is for g at least 2, and the low-genus cases (g = 0, 1) are genuine exceptions, not afterthoughts. State the hypothesis, never the slogan. Second, this guide is a survey: Abel's theorem we stated and motivated but did not prove (the proof runs through the Riemann bilinear relations and reciprocity), and theta functions, the Torelli theorem (that a curve is recoverable from its polarized Jacobian), and the Schottky problem (which abelian varieties are Jacobians) are all named here but unproved. A real proof of any of them is a chapter or a course, and pretending otherwise would be dishonest.
Finally, conventions: the period matrix, the polarization, and the theta function all carry sign and normalization choices that differ across Griffiths-Harris, Farkas-Kra, and Mumford; some authors put the lattice as columns, others as rows; the factor of 2 pi i lurks in different places. Pick one source, write its conventions inside the cover, and stay loyal — the same discipline guide 1 urged. The moduli dimension 3g - 3, the Jacobian dimension g, and Abel's u(D) = 0 are convention-independent and safe to memorize; the explicit matrices are not.
Look back across the whole rung. A compact Riemann surface arrived as topology, analysis, and algebra fused (guide 1); maps between surfaces obeyed Riemann-Hurwitz (guide 2); divisors and the canonical class were counted by Riemann-Roch (guide 3); uniformization sorted the three geometries (guide 4); and now the Abel-Jacobi map has linearized divisor classes into the Jacobian, with moduli space organizing all surfaces of fixed genus. From here the ladder forks upward: into the higher-dimensional complex geometry of Kahler manifolds and Hodge theory, into the scheme-theoretic algebraic geometry where curves become one chapter of a vast edifice, and into the mapping-class-group and Teichmuller theory that touches hyperbolic 3-manifolds and physics. You now hold the genus-1 picture firmly enough that each of those next floors will read as a generalization rather than a mystery.