Theta Functions for Polarized Calabi-Yau Varieties

本研究的主要目的是将经典的Abelian变种的theta函数理论推广到极化的Calabi-Yau变种，既开放（即log）又紧致，更精确地说：给出全局截面的向量空间的正则基，以及坐标环中乘法规则的结构常数的公式，用正则基表示，由镜像上的有理曲线的计数决定。这些广义函数的存在表明模空间存在几何上有意义的紧化，极大地推广了A_{g，d}的mumford - namikava - alexeev - olsson紧化和二次多面体的Gelfand-Kapranov-Zelevinski-Alexeev-Olsson理论，同时提出了作为正则描述环的投影的镜像的合成构造。该建议包括在维度2中执行此操作的详细方案，以及所有维度的集群品种。主要的建议是，一类广泛的几何物体，即所谓的Calabi-Yau品种，具有自然的坐标系。非正式地说：如果你生活在卡拉比-丘的品种上，你的世界里应该有一些自然的、内在的量，它们的值决定了你的精确位置。由于这些几何对象在数学的各个领域中发挥着基础作用，这些内在量也应该发挥类似的基础作用。更广泛地说，弦理论模型表明我们生活在一个卡拉比-丘上，因此这个提议表明，在我们的世界中存在这样一些尚未被理解的基本量。

The main objective of the proposed research is to generalize the classical theory of theta functions for Abelian varieties to polarized Calabi-Yau varieties, both open (i.e. log) and compact, more precisely: to give a canonical basis for the vector space of global sections, and a formula for the structure constants for the multiplication rule in the coordinate ring, expressed in the canonical basis, determined by counts of rational curves on the mirror. The existence of such generalized theta functions points to the existence of geometrically meaningful compactification of the moduli space, vastly generalizing the Mumford-Namikawa-Alexeev-Olsson compactificaton of A_{g,d}, and the Gelfand-Kapranov-Zelevinski-Alexeev-Olsson theory of the secondary polytope, and at the same time suggests a synthetic construction of the mirror as Proj of the canonically described ring. The proposal includes a detailed scheme for carrying this out in dimension two, and for cluster varieties of all dimensions.The main proposal is that a broad class of geometric objects, so called Calabi-Yau varieties, come with a natural system of coordinates. Informally: If you live on a Calabi-Yau variety, there should be natural, intrinsic quantities in your world, whose values determine your precise position. As these geometric objects play a fundamental role in diverse areas of mathematics, these intrinsic quantities should play a similar fundamental role. More broadly, string theory models suggest that WE live on a Calabi-Yau, and thus the proposal suggests there are such fundamental quantities, not yet understood, in our world.