AI 中文总结
该研究针对环面三维簇中的椭圆曲线，通过将均匀间距亏格1热带曲线与对数退化公式结合，得到Getzler–Pandharipande公式的对数类似物，证明ℙ³的对数虚拟不变量次数足够大时小于普通格罗莫夫-威滕不变量。
AI 中文摘要
我们研究环面三维簇中椭圆曲线的枚举几何。我们考虑称为均匀间距计数的枚举整数不变量，这类不变量可通过ℝ³中的均匀间距亏格1热带曲线进行研究。将其与对数退化公式对比，我们得到对数虚拟不变量与这些几何不变量之间的明确关系。该结果是Getzler–Pandharipande公式针对ℙ³中椭圆曲线的对数类似物。作为应用，我们证明，当次数足够大时，ℙ³相对于其环面边界的对数虚拟不变量严格小于普通格罗莫夫-威滕不变量。文中包含若干例子。
英文摘要
We study the enumerative geometry of elliptic curves in toric threefolds. We consider enumerative integer invariants, called well-spaced counts, which can be studied using well-spaced genus-one tropical curves in $\mathbb{R}^3$. By comparing this with the logarithmic degeneration formula, we obtain an explicit relationship between logarithmic virtual invariants and these geometric invariants. The result is a logarithmic analogue of a formula of Getzler--Pandharipande for elliptic curves in $\mathbb{P}^3$. As an application, we show that the virtual logarithmic invariants for $\mathbb{P}^3$ with respect to its toric boundary are strictly less than the ordinary Gromov--Witten invariants once the degree is sufficiently large. Several examples are included.
Comments36 pages, 19 figures. Comments welcome! v2: minor corrections and clarifications
