Irreducible cone spherical metrics and stable extensions of two line bundles
Irreducible cone spherical metrics and stable extensions of two line bundles
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DOI：
10.1016/j.aim.2021.107854
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发表时间：
2020-01
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影响因子：
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通讯作者：
Lingguang Li;Jijian Song;Bin Xu
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文献类型：
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作者：
Lingguang Li;Jijian Song;Bin Xu
一个锥球度量称为不可约的，如果该度量的任何展开映射在U（1）中不具有单值性。利用固有丛理论，在亏格为gX ≥ 1的紧致Riemann曲面X上构造了一个从两个线丛的稳定扩张的模空间到锥角在2 π Z> 1内的不可约度量的模空间的典型满射，当gX ≥ 2时，该满射在代数几何意义上是一般内射的.作为应用，我们证明了以下关于不可约度量的两个结果：·当g X≥ 2且d为偶数且大于12 g X-7时，可用不可约度量表示的d次有效因子形成Sym d（X）中Hausdorff维度≥ 2（d+ 3 - 3 g X）的弧连通Borel子集;·当g X≥ 1时，对于X上几乎所有次数为奇数且大于2 g X-2的有效因子D，存在有限多个表示D的锥球度量。
A cone spherical metric is called irreducible if any developing map of the metric does not have monodromy in U (1). By using the theory of indigenous bundles, we construct on a compact Riemann surface X of genus g X≥ 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 Z> 1, which is generically injective in the algebro-geometric sense as g X≥ 2. As an application, we prove the following two results about irreducible metrics:• as g X≥ 2 and d is even and greater than 12 g X− 7, the effective divisors of degree d which could be represented by irreducible metrics form an arcwise connected Borel subset of Hausdorff dimension≥ 2 (d+ 3− 3 g X) in Sym d (X);• as g X≥ 1, for almost every effective divisor D of degree odd and greater than 2 g X− 2 on X, there exist finitely many cone spherical metrics representing D.