Calculation of Gross-Siebert mirror rings for log Calabi-Yau manifolds.

对数 Calabi-Yau 流形的 Gross-Siebert 镜环的计算。

• 批准号: 2279765
• 金额: --
• 依托单位: University of Cambridge
• 依托单位国家: 英国
• 项目类别: Studentship
• 财政年份: 2019
• 资助国家: 英国
• 起止时间: 2019 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-2279765
• 关键词: Calculation; Gross; Siebert; mirror; rings

Mirror symmetry, first introduced by string theorists around 1989, posits a duality between two different geometric objects. Having now had a broad impact on algebraic, differential and symplecticgeometry, mirror symmetry is a very broad and active field in mathematics. Gross and Siebert have recently developed a theoretical approach to construction of mirrors in great generality, but to datethere have been very few explicit calculations using this approach.Wang's project aims to develop a suite of tools for calculation of the Gross-Siebert mirrors to log Calabi-Yau manifolds. Wang will initially consider the ring of functions associated to a pair ofa Fano variety with a smooth anti-canonical divisor as boundary. This will lead to new formulas involving rational curves meeting the anti-canonical divisor at two points. It will also makeconnection to recent work of Fan, Wu and You, who have constructed a similar theory using orbifold invariants.After developing a satisfactory understanding when the boundary is smooth, Wang will extend these techniques to new examples with a normal crossings boundary. This will lead to new constructions of mirror pairs and greatly enhance our understanding of the Gross-Siebert mirror construction.

镜像对称，最早是由弦理论家在1989年左右提出的，它假定了两个不同几何物体之间的对偶性。镜像对称对代数、微分和辛几何产生了广泛的影响，是数学中一个非常广泛和活跃的领域。Gross和Siebert最近开发了一种理论方法来构造反射镜，但到目前为止，很少有明确的计算使用这种方法。Wang的项目旨在开发一套工具来计算Gross-Siebert反射镜以记录Calabi-Yau流形。王将首先考虑与一对Fano簇相关的函数环，其边界为光滑的反典型因子。这将导致新的公式，涉及有理曲线满足反典型因子在两个点。它也将makeconnection到最近的工作范，吴和你，谁建立了一个类似的理论使用orbifold不变量。发展一个令人满意的理解后，边界是光滑的，王将这些技术扩展到新的例子与正常的交叉边界。这将导致新的结构的镜对，大大提高了我们的理解Gross-Siebert镜建设。