解析的捩率の幾何学的性質とその応用に関する研究

BCOV不变量是卡-丘流形的不变量。证明了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。这位联合研究者预计，他的研究可以激活这些属于截然不同的数学分支的研究领域之间的新互动。

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.