解析的捩率の幾何学的性質とその応用に関する研究

BCOV不变量是Calabi-Yau流形的一个不变量。有人猜想BCOV不变量是一个二元不变量。合作调查员的最终目标是证明这一猜想。该方法基于如下定理：代数簇之间的任何二次等价都可以分解为具有光滑中心的Blow-up/Blow-down。研究计划包括两个步骤1。我们构造了带有简单正规交叉正则因子的Kaehler流形的BCOV不变量，这并不一定有效。我们研究了扩展的BCOV不变量在爆破下的行为.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.第二步：爆破下BCOV不变量的变化由爆破中心的拓扑类型唯一地决定。这一结果暗示了猜想的一个弱版本：BCOV不变量是模为形变类型不变量的二元不变量。这位合作调查员还表明，这个猜想也适用于Mukai的失败。在上述研究中，合作研究者基本上使用了双曲面几何和Quillen度量学理论的结果。这位合作研究人员预计，他的研究可以激活这些研究领域之间的新互动，这些领域属于非常不同的数学分支。

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.