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Table of contents

Then Z Z 9. RZ In the proof theorem 9. Definition The set of P all these is denoted by Div S. Denote by Prin S the set of all principal divisors. Group structure of Div S. The set Div S is really the free abelian group generated by S. We denote the subset of divisors of degree d by Divd S.

By corollary 8. Example Linear equivalence. Theorem Below we will show that as an abelian group, Pic S only depends on the genus of S. Canonical class. There is a particularly important linear equivalence class, on each compact Riemann surface S, determined by the meromorphic differentials. By lemma 8. From lemma 8. Similarly, every divisor in the class [K ] is the divisor of zeroes and poles of some meromorphic differential. We call the class [K ] the canonical class of S. So we have proved Lemma Integrals modulo periods.

Hence we must work with integrals modulo periods. Jacobi variety.

Families Of Meromorphic Functions On Compact Riemann Surfaces

Jac S has several compatible structures: 1. Complex manifolds with all these properties in fact 2 implies 3 are called abelian varieties. Examples of Jacobi varieties. Remark From now on we will think of Jac S this way. In particular, its complex structure is the quotient structure for the covering map This perspective allows us to prove a useful result. Lemma The Abel map. We also fix chart V , w about AP Q as follows. Therefore dF We want to generalise the Abel map to positive divisors.

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Clearly S d is the moduli space of positive divisors of degree d on S. About each Pj choose a chart Uj , zj on S centred at Pj. This is an injective map, since each divisor is uniquely determined by the zeroes of the polynomial in It also has open image. It is an exercise to show that when two such charts overlap the change of coordinates is holomorphic.

Bibliographic Information

Abel map on divisors. To generalise the Abel map to positive divisors, first we make an observation about so-called Abelian sums. The Abel map on divisors is holomorphic. The proof will be given later, at which point we will also be able to prove the Riemann-Roch theorem. Before we begin the proof we need an intermediate lemma.

This follows from the fact that if z1 ,. By the previous lemma and theorem This is a meromorphic differential with the properties: 1. Then we have the following result. Proof of 1. Proof of 2. The differential of the Abel map. The aim of the Riemann-Roch theorem is to compute this dimension. This dimension is the kernel dimension of the differential To begin, we compute this differential in coordinates. The Brill-Noether matrix. We can use some of the calculations from the proof of lemma Therefore, as in the proof of lemma The Riemann-Roch theorem.

Recall that [K ] is the canonical class: the linear equivalence class of divisors for meromorphic differentials. The fascinating thing about this theorem is that the information on the left is holomorphic data while the information on the right is topological data.

Proof of Riemann-Roch. Proof of Riemann-Roch contd. So we have proved Now, since h0 D depends only on [D], we have proved We now prove the claim. We need another lemma. Proof of lemma. We can apply To prove the claim, choose any basis f1 ,.


Note that if Qj appears multiply in this divisor the linear equations involve derivatives of the fm evaluated at Qj. But in that case Some remarks about Riemann-Roch: 1. Some remarks about Riemann-Roch: 2. The Brill-Noether matrix plays a major role in this study. Some remarks about Riemann-Roch: 3 The modern proofs of Riemann-Roch use sheaf theory and consequently can be extended in much greater generality to apply to divisor classes via line bundles for compact complex manifolds of any dimension.

However, the analytic proof uses tools the exponential function not available to algebraic geometry, hence there are two approaches, analytic and algebraic, which look very different. Each proof has its virtues: the analytic proof for higher dimensions works on non-algebraic manifolds, while the algebraic proof works on non-manifold varieties varieties with singularities and over ground fields with non-zero characteristic. The Picard group and the Jacobian. Proof of First we show this is well-defined. But this forces the image to equal Jac S.

Meromorphic Functions

It is a deep theorem the Torelli theorem that this structure uniquely determines S. Embeddings in CPn. Fix a compact Riemann surface S of genus g and consider a positive divisor D on S of degree d. By lemma We can attempt to map S holomorphically into CPn as follows. Fix a basis f0 ,. Notice that the choice of basis only alters the map up to congruence in CPn , i.

Immersions and embeddings. It gives a meromorphic function in each chart, hence a globally meromorphic function f. Otherwise there would be an accumulation of zeroes or poles in some chart.

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The residue theorem Theorem 8. P Proof.

Corollary 8. Then X X 8. Hence, by the residue theorem, we obtain 8. The previous argument also provides us with a useful lemma concerning local coordinates about a branch point of a meromorphic function. Choose U to be a connected and simply connected neighbourhood of P in which f has no zero other than P and let w be a coordinate in U centred at P.

About this book

Period integrals. Integrals of holomorphic or meromorphic differentials around closed paths are called period integrals or periods.

It is a major theorem the Torelli theorem, beyond the scope of the course that the period integrals tell us everything about the Riemann surface. The Period lattice. Lemma 9. Corollary 9. One of the crucial facts required for understanding meromorphic function theory on a compact Riemann surface is the existence of certain meromorphic differentials, classically called differentials of the third kind. Theorem 9. Proof of 9. The formula 9.