论文标题
毛i的内在镜子环，用于光滑的log calabi-yau对
Gross-Siebert intrinsic mirror ring for smooth log Calabi-Yau pairs
论文作者
论文摘要
在本文中，我们展示了Gross和Siebert使用的刺穿的Gromov-witten不变性的公式，与一个点相关的Gromov-Witten不变性剂和一个点构成，用于任何光滑的log log log calabi-yau Pair $（W，d）$。用$ n_ {a，b} $表示$ w $ secilet $ d $的合理曲线数量为两个点，一个带有联系订单$ a $的$ d $，一个带有点约束的联系订单$ b $。 （此类数字是在相对或对数Gromov-witten理论中定义的）。 We then apply a modified version of deformation to the normal cone technique and the degeneration formula developed by Kim, Lho, Ruddat and Abramovich, Chen, Gross, Siebert to give a full understanding of $N_{e-1,1}$ with $D$ nef where $e$ is the intersection number of $D$ and a chosen curve class.后来，通过刺穿的不变性作为辅助不变性，我们证明，对于带有椭圆曲线$（\ Mathbb {p}^2，d）$的投射平面，所有标准的2点$ d $，具有一个$ d $的相对不变性，每个$ d $，可以由这些$ d $确定的n $ d $ d $ d $ d $ d nam $ n_ {3d-1,1} $，加上那些低度不变的。在最后一节中，我们为$（\ Mathbb {p}^2，d）$的两点，度2分2点，单点的相对相对不变式进行了完整计算。
In this paper, we exhibit a formula relating punctured Gromov-Witten invariants used by Gross and Siebert to 2-point relative/logarithmic Gromov-Witten invariants with one point-constraint for any smooth log Calabi-Yau pair $(W,D)$. Denote by $N_{a,b}$ the number of rational curves in $W$ meeting $D$ in two points, one with contact order $a$ and one with contact order $b$ with a point constraint. (Such numbers are defined within relative or logarithmic Gromov-Witten theory). We then apply a modified version of deformation to the normal cone technique and the degeneration formula developed by Kim, Lho, Ruddat and Abramovich, Chen, Gross, Siebert to give a full understanding of $N_{e-1,1}$ with $D$ nef where $e$ is the intersection number of $D$ and a chosen curve class. Later, by means of punctured invariants as auxiliary invariants, we prove, for the projective plane with an elliptic curve $(\mathbb{P}^2, D)$, that all standard 2-pointed, degree $d$, relative invariants with a point condition, for each $d$, can be determined by exactly one of these degree $d$ invariants, namely $N_{3d-1,1}$, plus those lower degree invariants. In the last section, we give full calculations of 2-pointed, degree 2, one-point-constrained relative Gromov-Witten invariants for $(\mathbb{P}^2, D)$.
