A criterion for Jacobi varieties. (English) Zbl 0574.14027

Welters, G. E.

Ann. Math. (2) 120, 497-504 (1984).

R. C. Gunning [Invent. Math. 66, 377-389 (1982; Zbl 0485.14009)] gives a criterion for a principally polarized abelian variety X to be the jacobian variety of a curve by means of the trisecants of the Kummer- Wirtinger variety of X. In a previous paper [Indagationes Math. 45, 501- 520 (1983; Zbl 0542.14029)], the author gives an infinitesimalization of Gunning’s criterion, and he improves his criterion in this paper. His results give a new view point to the so-called Schottky problem. In particular, E. Arbarello and C. De Concini [Ann. Math., II. Ser. 120, 119-140 (1984; Zbl 0551.14016)] give an analytic translation of his results, and show the existence of a close relationship between their criterion and the Novikov conjecture.

Reviewer: T.Sekiguchi

MSC:

14H40	Jacobians, Prym varieties
14K25	Theta functions and abelian varieties
30F10	Compact Riemann surfaces and uniformization
14K30	Picard schemes, higher Jacobians

Keywords:

jacobian variety; trisecants of the Kummer-Wirtinger variety; Schottky problem

Citations:

Zbl 0485.14009; Zbl 0542.14029; Zbl 0551.14016