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流形的不变量。我们推测BCOV不变量是一个双分不变量。合作研究者的最终目标是证明这一猜想。该方法基于以下定理：代数变量之间的任何双分等价都可以分解为中心光滑的blow-up/blow-down。研究计划包括两个步骤。我们构造了具有简单正规交叉正则除数的Kaehler流形的BCOV不变量，这种不变量不一定有效。研究了膨胀条件下扩展BCOV不变量的行为。合作者完成了第1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。合作者在第二步上取得了相当大的进展：爆破下BCOV不变量的变化是唯一由爆破中心的拓扑类型决定的。这个结果暗示了这个猜想的一个弱版本：BCOV不变量是一个双分不变量模一个变形型不变量。这位共同研究者还证明了这一猜想适用于Mukai flops。在上面提到的研究中，合作研究者基本上使用了双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的互动，这些研究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.