Irreducible cone spherical metrics and stable extensions of two line bundles

• Published in: arXiv.org 2021-10
• Main Authors: Li, Lingguang, Song, Jijian, Xu, Bin
• Format: Article
• Language: English
• Subjects: Angles (geometry); Bundles; Bundling; Riemann surfaces
• ISSN: 2331-8422
• DOI: 10.48550/arxiv.2001.08872

A cone spherical metric is called irreducible if any developing map of the metric does not have monodromy in \({\rm U(1)}\). By using the theory of indigenous bundles, we construct on a compact Riemann surface \(X\) of genus \(g_X \geq 1\) a canonical surjective map from the moduli space of stable extensions of two line bundles to that of irreducible metrics with cone angles in \(2 \pi \mathbb{Z}_{>1}\), which is generically injective in the algebro-geometric sense as \(g_X \geq 2\). As an application, we prove the following two results about irreducible metrics: \(\bullet\) as \(g_X \geq 2\) and \(d\) is even and greater than \(12g_X - 7\), the effective divisors of degree \(d\) which could be represented by irreducible metrics form an arcwise connected Borel subset of Hausdorff dimension \(\geq 2(d+3-3g_X)\) in \({\rm Sym}^d(X)\); \(\bullet\) as \(g_X \geq 1\), for almost every effective divisor \(D\) of degree odd and greater than \(2g_X-2\) on \(X\), there exist finitely many cone spherical metrics representing \(D\).