解析的捩率の幾何学的性質とその応用に関する研究

解析扭转的几何性质及其应用研究

• 批准号: 17F17804
• 负责人: 吉川 謙一
• 金额: $ 0.9万
• 依托单位: Kyoto University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for JSPS Fellows
• 财政年份: 2017
• 资助国家: 日本
• 起止时间: 2017-11-10 至 2020-03-31
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-17F17804/
• 关键词: Analytic torsion; BCOV invariant; Calabi-Yau manifolds; birational maps; analytic torsion; scattering theory

The BCOV invariant is an invariant for Calabi-Yau manifolds. It was conjectured that the BCOV invariant is a birational invariant. The co-investigator's ultimate goal is to prove this conjecture. The approach is based on the following theorem: any birational equivalence between algebraic varieties can be decomposed into blow-up/blow-down with smooth center. The research program consists of two steps.1. We construct BCOV invariant for Kaehler manifolds equipped with a simple normal crossing canonical divisor, which is not necessarily effective.2. We study the behavior of the extended BCOV invariant under blow-up.The co-investigator accomplished step 1 and showed that for rigid del Pezzo surfaces, the invariant obtained is equivalent to Yoshikawa's equivariant BCOV invariant. The co-investigator obtained considerable progress on step 2: the change of BCOV invariant under blow-up is uniquely determined by the topological type of the center of the blow-up. This result implies a weak version of the conjecture: the BCOV invariant is a birational invariant modulo a deformation type invariant. The co-investigator also showed that the conjecture holds for Mukai flops. In the research mentioned above, the co-investigator essentially used results from birational geometry and the theory of Quillen metrics. The co-investigator expects that his research could activate new interaction between these research areas, which belong to very different branches of mathematics.

BCOV不变量是卡-丘流形的不变量。据推测，BCOV不变量是一个双有理不变量。共同研究者的最终目标是证明这个猜想。该方法基于以下定理：代数簇之间的任何双有理等价都可以分解为具有光滑中心的blow-up/blow-down。该研究计划包括两个步骤。1.我们构造了具有一个简单法向交叉标准因子的Kaehler流形的BCOV不变量，这不一定是有效的.我们研究了扩展BCOV不变量在Blow-up下的行为，合作研究者完成了第一步，并证明了对于刚性del Pezzo曲面，所得到的不变量等价于Yoshikawa的等变BCOV不变量。合作研究者在第二步取得了相当大的进展：BCOV不变量在爆破下的变化唯一地由爆破中心的拓扑类型决定。这个结果暗示了一个弱版本的猜想：BCOV不变量是一个双有理不变量模的变形型不变量。合作研究者还表明，该猜想适用于向井的失败。在上面提到的研究中，合作研究者基本上使用了双有理几何和奎伦度量理论的结果。共同研究者希望他的研究能够激活这些研究领域之间的新互动，这些研究领域属于非常不同的数学分支。

项目成果

Adiabatic limit, Witten deformation and analytic torsion forms

绝热极限、维滕变形和解析扭转形式

• 发表时间: 2018
• 作者: Cecile Laly;ZURC,Joca;Yeping Zhang
• 通讯作者: Yeping Zhang

Scattering matrix and analytic torsion

散射矩阵和解析扭转

• 期刊: Analysis & PDE
• 影响因子: 2.2
• 作者: Martin Puchol;Yeping Zhang;Jialin Zhu
• 通讯作者: Jialin Zhu

BCOV invariant and ramified cover

BCOV 不变和分支覆盖

• 发表时间: 2019
• 作者: Maciej Krzeszewski;Takehisa Maekawa;Hideto Ito;Kenichiro Itami;ラリ・セシル;Yeping Zhang
• 通讯作者: Yeping Zhang

The direct image of a flat fibration with complex fibers

具有复杂纤维的扁平纤维的直接图像

• 发表时间: 2020
• 期刊: Bulletin des Sciences Mathematiques
• 影响因子: 1.3
• 作者: Yeping Zhang

BCOV invariant and birational equivalence

BCOV 不变性和双有理等价

• 发表时间: 2019
• 作者: Zurc;J.;Laly Cecile;Yeping Zhang
• 通讯作者: Yeping Zhang