$L^2$-Riemann-Roch for singular complex curves

TL;DR

This paper develops an $L^2$-theory for the $ar{ ext{d}}$-operator on singular complex curves, extending classical results like Riemann-Roch to singular settings and exploring their applications.

Contribution

It introduces an $L^2$-framework for the $ar{ ext{d}}$-operator on singular curves, including an $L^2$-version of the Riemann-Roch theorem, which is novel in this context.

Findings

Establishes an $L^2$-theory for the $ar{ ext{d}}$-operator on singular curves.

Derives an $L^2$-version of the Riemann-Roch theorem.

Provides applications of the $L^2$-theory to complex geometry.

Abstract

We present a comprehensive $L^2$-theory for the $\overline\partial$-operator on singular complex curves, including $L^2$-versions of the Riemann-Roch theorem and some applications.