论文标题
两条线束的不可总体球形指标和稳定的扩展
Irreducible cone spherical metrics and stable extensions of two line bundles
论文作者
论文摘要
锥形球形指标被称为不可修复，如果任何开发的度量图都没有$ {\ rm u（1）} $中的单片。通过使用本地捆绑包的理论，我们在紧凑的Riemann表面上构建$ x $ $ g_x \ geq 1 $从模量稳定的稳定范围的规范溢流图，两行套件的稳定扩展空间，以$2π\ Mathbb {Z} $ niff in Is niff cone Angive cone Angive of Cone Ange的稳定扩展。代数 - 几何感作为$ g_x \ geq 2 $。作为应用程序，我们证明了以下两个有关不可约指标的结果： $ \ bullet $ as $ g_x \ geq 2 $和$ d $均匀且大于$ 12G_X -7 $，是$ d $的有效分配，可以由不可约定的度量标准表示，形成了弧形连接的弧形borel borel子集的Hausdorff dimension $ \ geq 2（d+3-3g_x）$ $ \ bullet $作为$ g_x \ geq 1 $，对于几乎每个有效的除数$ d $ of ofd ofd ofd ofd ofd ofd $ d $ d $ d $ d $ d $ x $上的$ 2G_x-2 $，有限许多代表$ d $的锥形球形指标。
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 $2 π\mathbb{Z}_{>1}$, 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 $D$.
