Mirror Symmetry for Stable Quotients Invariants
Mirror Symmetry for Stable Quotients Invariants
复制标题
稳定商不变量的镜像对称
DOI：
10.1307/mmj/1409932634
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发表时间：
2012
期刊：
影响因子：
--
通讯作者：
A. Zinger
中科院分区：
文献类型：
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作者：
Y. Cooper;A. Zinger
Marian-Oprea-Pandharipande 引入的稳定商模空间提供了从非奇异曲线到非奇异射影簇的态射空间的自然紧化，并带有自然的虚类。我们证明了所得到的扭曲射影不变量的吉文塔尔 J 函数的类似物是由与相应的 Gromov-Witten 不变量（来自稳定映射的模空间）相同的镜像超几何级数描述的，但没有镜像变换（在 Calabi-Yau 情况下）。这意味着如果有足够的“正性”，稳定商和 Gromov-Witten 扭曲不变量会一致，但并非在所有情况下都是一致的。作为证明的推论，我们表明稳定商理论中出现的某些扭曲赫尔维茨数也可以由与该超几何级数相关的基本对象来描述。因此，我们完全回答了 Marian-Oprea-Pandharipande 提出的关于它们的不变量的一些问题。我们的结果表明完全相交的稳定商不变量与镜子族的几何形状之间存在深刻的联系。与 Gromov-Witten 理论一样，计算 Givental 的 J 函数（本质上是具有 1 个标记点的属 0 不变量的生成函数）是计算更高属和具有更多标记点的稳定商不变量的关键；我们将在即将发表的论文中利用这一点。
The moduli space of stable quotients introduced by Marian-Oprea-Pandharipande provides a natural compactification of the space of morphisms from nonsingular curves to a nonsingular projective variety and carries a natural virtual class. We show that the analogue of Givental's J-function for the resulting twisted projective invariants is described by the same mirror hypergeometric series as the corresponding Gromov-Witten invariants (which arise from the moduli space of stable maps), but without the mirror transform (in the Calabi-Yau case). This implies that the stable quotients and Gromov-Witten twisted invariants agree if there is enough "positivity", but not in all cases. As a corollary of the proof, we show that certain twisted Hurwitz numbers arising in the stable quotients theory are also described by a fundamental object associated with this hypergeometric series. We thus completely answer some of the questions posed by Marian-Oprea-Pandharipande concerning their invariants. Our results suggest a deep connection between the stable quotients invariants of complete intersections and the geometry of the mirror families. As in Gromov-Witten theory, computing Givental's J-function (essentially a generating function for genus 0 invariants with 1 marked point) is key to computing stable quotients invariants of higher genus and with more marked points; we exploit this in forthcoming papers.