Theta Functions for Polarized Calabi-Yau Varieties

这项研究的主要目的是将经典的Abelian簇的theta函数理论推广到极化的Calabi-Yau簇，包括开的(即对数的)和紧的，更准确地说：给出全局截面的向量空间的规范基，以及由镜面上有理曲线的个数决定的坐标环中乘法规则的结构常数的结构常数的公式，用标准基表示。这种广义theta函数的存在表明了模空间的几何意义紧化的存在，极大地推广了A_{g，d}的Mumford-纳米kawa-Alexeev-Olsson紧化和第二多面体的Gelfand-Kapranov-Zlevinski-Alexeev-Olsson理论，同时提出了作为正则描述的环的投影的镜面的综合构造.该方案包括在第二维空间和所有维度的星团变种中实现这一点的详细方案。主要方案是，一大类几何物体，即所谓的Calabi-Yau变种，具有一个自然的坐标系。非正式：如果你生活在Calabi-Yau品种上，你的世界应该有自然的、内在的数量，它们的值决定了你的准确位置。由于这些几何对象在不同的数学领域中扮演着基本的角色，这些内在的量也应该扮演类似的基本角色。更广泛地说，弦理论模型表明，我们生活在卡拉比-尤行星上，因此，该提议表明，在我们的世界中存在这样的基本量，但尚不清楚。

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.