AI 中文总结
本文为系列研究首篇，将椭圆曲线格罗莫夫-维滕理论拟模性推广至高维，经重言投影后证明亏格2情形的相关猜想，引入商格罗莫夫-维滕不变量并给出其显式公式。
AI 中文摘要
本文是关于 h 维主极化阿贝尔簇模空间上通用阿贝尔簇的格罗莫夫-维滕理论系列研究的首篇论文。我们提出猜想：当对主极化的次数求和时，格罗莫夫-维滕类的生成级数是 SL₂(ℤ) 的取值于闭链的拟模形式，且满足全纯反常方程。这些猜想将椭圆曲线格罗莫夫-维滕理论的拟模性推广到更高维度，并引发了关于阿贝尔簇枚举镜像对称的有趣问题。在亏格 1 情形，该猜想退化为 Greer 与 Lian 的猜想，Iribar Lopez 经重言投影后已证明该猜想。我们还讨论了一类特殊的阿贝尔簇族，其与更高亏格的西格尔拟模形式存在猜想性关联。本文主要结果是经重言投影后证明了亏格 2 情形下的上述猜想。为此，我们引入了由曲线类的特征多项式索引的商格罗莫夫-维滕不变量，并证明其可确定所有满足次数条件的后代格罗莫夫-维滕不变量。随后，我们给出经重言投影后所有亏格 2 商不变量的显式公式，该公式为两个艾森斯坦级数乘积的志村提升，其推导基于与 Brandon Williams 合作附录中得到的一个奇特模恒等式。
英文摘要
This is the first paper in a series on the Gromov-Witten theory of the universal abelian variety over the moduli space of principally polarized abelian varieties of dimension $h$. We conjecture that the generating series of Gromov-Witten classes, when summed over the degree against the principal polarization, is a cycle-valued quasimodular form for $\mathrm{SL}_2(\mathbb{Z})$ and satisfy a holomorphic anomaly equation. These conjectures generalize the quasimodularity of the Gromov-Witten theory of elliptic curves to higher dimension and raise interesting questions regarding enumerative mirror symmetry for abelian varieties. In genus $1$ it specializes to a conjecture of Greer and Lian which was proven by Iribar Lopez after tautological projection. We also discuss a special family of abelian varieties with a conjectural relation to Siegel quasimodular forms of higher genus. The main result of the paper is a proof of the conjectures in genus $2$ after tautological projection. For that we introduce quotient Gromov-Witten invariants which are indexed by the characteristic polynomial of the curve class and are shown to determine all descendent Gromov-Witten invariants satisfying a degree conditions. We then give an explicit formula for all genus $2$ quotient invariants after tautological projection as the Shimura lift of the product of two Eisenstein series. The formula is based on a curious modular identity derived in a joint appendix with Brandon Williams.
Comments57 pages. Comments welcome
