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Lectures on Riemann Surfaces: Jacobi Varieties

Lectures on Riemann Surfaces: Jacobi Varieties

Copyright Date: 1972
Pages: 195
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  • Book Info
    Lectures on Riemann Surfaces: Jacobi Varieties
    Book Description:

    A sequel toLectures on Riemann Surfaces(Mathematical Notes, 1966), this volume continues the discussion of the dimensions of spaces of holomorphic cross-sections of complex line bundles over compact Riemann surfaces. Whereas the earlier treatment was limited to results obtainable chiefly by one-dimensional methods, the more detailed analysis presented here requires the use of various properties of Jacobi varieties and of symmetric products of Riemann surfaces, and so serves as a further introduction to these topics as well.

    The first chapter consists of a rather explicit description of a canonical basis for the Abelian differentials on a marked Riemann surface, and of the description of the canonical meromorphic differentials and the prime function of a marked Riemann surface. Chapter 2 treats Jacobi varieties of compact Riemann surfaces and various subvarieties that arise in determining the dimensions of spaces of holomorphic cross-sections of complex line bundles. In Chapter 3, the author discusses the relations between Jacobi varieties and symmetric products of Riemann surfaces relevant to the determination of dimensions of spaces of holomorphic cross-sections of complex line bundles. The final chapter derives Torelli's theorem following A. Weil, but in an analytical context.

    Originally published in 1973.

    ThePrinceton Legacy Libraryuses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These paperback editions preserve the original texts of these important books while presenting them in durable paperback editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.

    eISBN: 978-1-4008-7269-5
    Subjects: Mathematics

Table of Contents

  1. Front Matter
    (pp. None)
  2. Preface.
    (pp. i-iv)
    R. C. Gunning
  3. Table of Contents
    (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)