Irreducible cone spherical metrics and stable extensions of two line bundles

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 $D$.