Some special complex vector bundles over Jacobi varieties

Abstract

It is by now quite familiar that the r-fold symmetric product of a compact Riemann surface of genus g >0 is in a natural way a complex analytic projective bundle over the Jacobi variety of that surface whenever r>2g-1. The Chern classes of the associated complex analytic vector bundles were determined by Mattuck [11] and Macdonald [9]; another approach to these bundles was developed by Schwarzenberger [16-1, and further applications of these methods to some classical problems have also recently appeared [6,7]. The aim of the present paper is to discuss these results from a more analytical point of view, so that they may be more accessible to analysts interested in the study of compact Riemann surfaces, and to point out some additional analytical properties of these bundles. The approach which will be used is that of constructing functions which simultaneously provide explicit bases for the spaces of holomorphic sections of all complex analytic line bundles of Chern class r over the Riemann surface when these bundles are represented by suitable factors of automorphy, and of using these functions in turn to construct factors of automorphy representing the desired vector bundles over the Jacobi variety; these functions appear to be useful generalizations of the classical theta functions, and the more detailed investigation of their properties is an interesting subject by itself but will not be pursued further here. The advantage of this approach is that the results obtained follow quite easily from simple applications of the usual Riemann-Roch theorem for compact Riemann surfaces, so that except for the interpretation of some of the results very little algebro-geometrical or topological machinery is really needed.In outline, Section 1 is a review of the description of complex analytic line bundles, and in particular of topologically trivial complex analytic line bundles over compact Riemann surfaces, by factors of automorphy. The construction of the basis functions is contained in Section 2 (Theorem 1), and the application of these functions to the description of the complex analytic vector bundles of interest is contained in Section 3. The principal properties of these functions used in the present discussion 13 Inventiones math., Vol. 22

A Property of the Periods of a Prym Differential

“…Then F(du) = Ar(ci)(F(«) -Ad) for all d in D and all u in U. By the calculation in the proof of Theorem 21 in [1], jR w2 A wr = -J. w2F equals the above expression.…”

“…Some basic facts about this vector bundle can be found in [2]. A general reference for periods of Prym differentials is the book [1].Given any Prym differential w, let V be the largest open subset of 77 such that w has zero residues everywhere in V. Denote the image of V in S by 7.Let v be any fixed point of V and let t be its image in 7. For any path a in 7, beginning and ending at t, the a period Aa of w is defined by the integral Aa= -X(a)-lJ,w, where a is the unique path in U, lifting a, which begins at t and ends at a ■ t. The deck transformation a will henceforth be denoted by a.…”

“…Some basic facts about this vector bundle can be found in [2]. A general reference for periods of Prym differentials is the book [1].…”

“…Let 0\,T\, ..., ag, Tg be a standard set [1] of closed arcs, beginning and ending at t, that dissect S. Let R be the area in U bounded by the lifting h,Oi]---[rg,og]. Proposition 1.…”

Inversion of Abelian integrals

“…With thesç general remarks out of the way, we can state This theorem [2,4] may be proven using general methods of Grauert, but there is no known way to construct such a trivialization explicitly.…”