Irreducible cone spherical metrics and stable extensions of two line bundles

Abstract

A cone spherical metric is called irreducible if any developing map of the metric does not have monodromy in $\mathrm{U}(1)$. By using the theory of indigenous bundles, we construct on a compact Riemann surface $X$ of genus $g_X\ge 1$ 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 $g_X\ge 2$. As an application, we prove the following two results about irreducible metrics: $\bullet$ as $g_X\ge 2$ 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 $\ge 2(d+3-3g_X)$ in ${\mathrm{Sym}}^d(X)$; $\bullet$ as $g_X\ge 1$, 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 $D$.

This manuscript supersedes arXiv:1808.04106. 34 pages

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