# Lectures on Riemann Surfaces: Jacobi Varieties

R. C. GUNNING
Pages: 195
https://www.jstor.org/stable/j.ctt130hkjm

1. Front Matter
(pp. None)
2. Preface.
(pp. i-iv)
R. C. Gunning
(pp. v-vi)
4. §1. Marked Riemann surfaces and their canonical differentials.
(pp. 1-33)

(a) At several points in the more detailed study of Riemann surfaces, the explicit topological properties of surfaces play an important role; and it is convenient to have these properties established from the beginning of the discussion, to avoid the necessity of inserting topological digressions later. Since the universal covering space of a connected orientable surface of genus g > 0 is a cell, the fundamental group carries essentially all the topological properties of the surface; so it is also convenient to introduce from the beginning and to use systematically henceforth the representation of a Riemann surface in terms of its...

5. §2. Jacobi varieties and their distinguished subvarieties.
(pp. 34-71)

(a) Consider once again the canonical period matrix (I,Ω) of a marked Riemann surface M , where$\text{I}=\{\delta_{\text{j}}^{\text{i}}\}$is the g × g identity matrix and Ω = {ωij} is the g × g matrix whose entries ωij= ωi(Bj) are the periods of the canonical holomorphic Abelian differentials. The columns of this matrix can be viewed as a set of 2g vectors in ℂg, and they are linearly independent over the real numbers as a consequence of Riemann's inequality; hence these vectors generate a lattice subgroup ℒ ⊂ ℂg, such that the quotient space ℂg/ℒ is a...

6. §3. Jacobi varieties and symmetric products of Riemann surfaces
(pp. 72-140)

(a) The restriction of the Jacobi homomorphism to the set of positive divisors of degree r can be viewed as a complex analytic mapping φ: Μr→ J(M) ; and it is evident that this mapping is really independent of the order of the factors in the Cartesian product Μr. This suggests introducing the symmetric product Μ(r), which is defined to be the quotient space${{\text{M}}^{(\text{r})}}={{\text{M}}^{\text{r}}}/{{\cal{G}}_{\text{r}}}$of the compact complex analytic manifold Mrunder the natural action of the symmetric group${{\cal{G}}_{\text{r}}}$on r letters as the group of permutations of the factors in the Cartesian product Mr. That is...

7. §4. Intersections in Jacobi varieties and Torelli's theorem.
(pp. 141-176)

(a) Intersections of subvarieties of positive divisors in the Jacobi variety have frequently been considered in the preceding discussion, as for example in the formulas from Lemmas 1 and 2,${{\text{W}}_{\text{s}-\text{r}}}={{\text{W}}_{\text{s}}}\circleddash {{\text{W}}_{\text{r}}}=\bigcap\limits_{\text{u}\in {{\text{W}}_{\text{r}}}}{({{\text{W}}_{\text{s}}}-\text{u})},\quad \text{o}\leqq \text{r}\leqq \text{s}\leqq \text{g}-\text{1},$$\text{W}_{\text{r}}^{\nu }={{\text{W}}_{\text{r}-\nu +1}}\circleddash (-{{\text{W}}_{\nu -1}})=\bigcap\limits_{\text{u}\in {{\text{W}}_{\nu -1}}}{({{\text{W}}_{\text{r}-\nu +1}}+\text{u}),}\quad \nu \leqq \text{r}+1.$

These are infinite intersections, though, and it is of some interest also to examine analogous finite intersections, as indicated by the discussion in §3(c). The proper investigation of these intersections requires some notion of intersection multiplicity, either analytical or topological; but leaving such complications aside for a later treatment, it is still possible to derive a number of interesting and useful results merely involving intersections in the point...

8. Appendix. On conditions ensuring that $\text {W}_{\text r}^{2}\ne \phi$.
(pp. 177-187)
9. Index of symbols
(pp. 188-188)
10. Index
(pp. 189-189)