The moduli space of holomorphic chains of rank one over a compact Riemann surface

TL;DR

This paper characterizes the moduli space of rank one holomorphic chains over a compact Riemann surface as a fiber product of projective space bundles, computes its Euler characteristic, and explores stability conditions related to real parameters.

Contribution

It provides a geometric description of the moduli space of rank one holomorphic chains and analyzes stability via real parameters and Gm characters.

Findings

Moduli space is a fiber product of projective space bundles.

Euler characteristic of the moduli space is computed.

Stability depends on real vector parameters and Gm characters.

Abstract

A holomorphic chain on a compact Riemann surface is a tuple of vector bundles together with homomorphisms between them. We show that the moduli space of holomorphic chains of rank one is identified with a fiber product of projective space bundles. We compute the Euler characteristic of the moduli space. The stability of chains involves real vector parameters. We also show that the variation of parameters corresponds to the characters of Gm.