论文标题
计数曲线上的滑轮
Counting sheaves on curves
论文作者
论文摘要
We compute Joyce's (arXiv:2111.04694) enumerative invariants $[\mathcal{M}^{\mathrm{ss}}_{(r,d)}]_{\mathrm{inv}}$ for semistable rank $r$ degree $d$ coherent sheaves on a complex projective curve.这些不变的人是对半固定滑轮模量的基本类别的概括。我们将不变性表示为正规总和，这是将有限值分配给发散序列的一种方式，并且我们获得了不变性的明确表达式。 从这些不变的人中，可以在可半固定滑轮的模量上提取共同体配对。当$ r $和$ d $是企业时，Witten发现了此类配对的公式，并由Jeffrey和Kirwan证明。我们的结果为这个经典问题提供了新的观点，可以将其视为对$ r $和$ d $不是企业的情况的概括。
We compute Joyce's (arXiv:2111.04694) enumerative invariants $[\mathcal{M}^{\mathrm{ss}}_{(r,d)}]_{\mathrm{inv}}$ for semistable rank $r$ degree $d$ coherent sheaves on a complex projective curve. These invariants are a generalization of the fundamental class of the moduli of semistable sheaves. We express the invariants as a regularized sum, which is a way to assign finite values to divergent series, and we obtain explicit expressions for the invariants. From these invariants, one can extract cohomology pairings on the moduli of semistable sheaves. When $r$ and $d$ are coprime, formulae for such pairings were found by Witten and proved by Jeffrey and Kirwan. Our results provide a new point of view on this classical problem, and can be seen as a generalization of this to the case when $r$ and $d$ are not coprime.
