Analytic study of singular curves

TL;DR

This paper provides an analytic framework for studying singular algebraic curves, establishing duality theorems, and exploring their associated varieties, extending classical results from smooth to singular cases.

Contribution

It offers new analytic proofs for duality and Abel's theorem and extends the understanding of Picard, Albanese, and Jacobi varieties to singular curves.

Findings

Analytic proofs of Serre duality and Abel's theorem for singular curves.

Detailed analysis of Albanese and Jacobi varieties in the singular case.

Extension of the relation between meromorphic functions on curves and their Jacobians to singular curves.

Abstract

We study singular curves from analytic point of view. We give completely analytic proofs for the Serre duality and a generalized Abel's theorem. We also reconsider Picard varieties, Albanese varieties and generalized Jacobi varieties of singular curves analytically. We call an Albanese variety considered as a complex Lie group an analytic Albanese variety. We investigate them in detail. For a non-singular curve (a compact Riemann surface) $X$, there is the relation between the meromorphic function fields on $X$ and on its Jacobi variety $J(X)$. We try to extend this relation to the case of singular curves.