Abel-Jacobi theorem

TL;DR

The Abel-Jacobi theorem links divisors on Riemann surfaces to their Jacobian varieties, providing a fundamental connection in algebraic geometry that facilitates the embedding of surfaces into complex tori.

Contribution

This paper discusses the classical Abel-Jacobi theorem, emphasizing the role of divisors, the Picard group, and the embedding of Riemann surfaces into Jacobian varieties.

Findings

Establishes the identification of the Picard group via the Abel-Jacobi map.

Shows how Riemann surfaces of genus g embed into their Jacobian varieties.

Highlights the importance of divisors and meromorphic functions in the theorem.

Abstract

The Abel Jacobi theorem is an important result of algebraic geometry. The theory of divisors and the Riemann bilinear relations are fundamental to the developement of this result: if a point O is fixed in a Riemann compact surface X of genus g, the Abel Jaobi map identifies the Picard group: the quotient of divisors of a group of degree zero by the sub-group of divisors associated to meromorphic functions. The Riemann surface of genus g can be embedded in the Jacobian variety via the Abel-Jacobi. In fact, generally.the surface may be provided with an analytical structure.or algebraic varietie.