Connections for weighted projective lines

TL;DR

This paper introduces a notion of connections on coherent sheaves over weighted projective lines and shows an equivalence between certain categories of these sheaves and representations of deformed preprojective algebras.

Contribution

It defines connections on coherent sheaves in this setting and establishes a new categorical equivalence under specific conditions.

Findings

Connections on coherent sheaves are well-defined in this context.

Categories of sheaves with connections relate to deformed preprojective algebra representations.

The result generalizes known equivalences in algebraic geometry and representation theory.

Abstract

We introduce a notion of a connection on a coherent sheaf on a weighted projective line (in the sense of Geigle and Lenzing). Using a theorem of Huebner and Lenzing we show, under a mild hypothesis, that if one considers coherent sheaves equipped with such a connection, and one passes to the perpendicular category to a nonzero vector bundle without self-extensions, then the resulting category is equivalent to the category of representations of a deformed preprojective algebra.