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Genus 0 Gopakumar–Vafa Invariants of the Banana Manifold
Quarterly Journal of Mathematics ( IF 0.7 ) Pub Date : 2021-04-30 , DOI: 10.1093/qmath/haab026
Nina Morishige 1
Affiliation  

The Banana manifold $X_{{\text{Ban}}}$ is a compact Calabi–Yau threefold constructed as the conifold resolution of the fiber product of a generic rational elliptic surface with itself, which was first studied by Bryan. We compute Katz’s genus 0 Gopakumar–Vafa invariants of fiber curve classes on the Banana manifold $X_{{\text{Ban}}}\to \mathbf{P} ^1$. The weak Jacobi form of weight −2 and index 1 is the associated generating function for these genus 0 Gopakumar–Vafa invariants. The invariants are shown to be an actual count of structure sheaves of certain possibly non-reduced genus 0 curves on the universal cover of the singular fibers of $X_{{\text{Ban}}}\to\mathbf{P}^1$.

中文翻译:

香蕉流形的第 0 类 Gopakumar–Vafa 不变量

Banana 流形 $X_{{\text{Ban}}}$ 是一个紧凑的 Calabi-Yau 三重结构,它是由 Bryan 首次研究的通用有理椭圆表面的纤维乘积与其自身的 conifold 分辨率。我们计算香蕉流形 $X_{{\text{Ban}}}\to \mathbf{P} ^1$ 上纤维曲线类的 Katz 属 0 Gopakumar–Vafa 不变量。权重 -2 和指数 1 的弱 Jacobi 形式是这些属 0 Gopakumar-Vafa 不变量的相关生成函数。不变量显示为 $X_{{\text{Ban}}}\to\mathbf{P}^1 的奇异纤维的普遍覆盖上某些可能非约简 0 类曲线的结构滑轮的实际计数美元。
更新日期:2021-04-30
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