Theta Functions for Polarized Calabi-Yau Varieties

提出的研究的主要目标是推广经典理论的theta函数的阿贝尔品种极化Calabi-Yau品种，都开放（即log）和compact，更确切地说：给出了整体截面向量空间的一个标准基，以及用标准基表示的坐标环中乘法规则的结构常数的公式，由镜子上的有理曲线的数量决定。这类广义theta函数的存在性表明模空间存在几何意义的紧化，极大地推广了A_{g，d}的Mumford-Namikawa-Alexeev-Olsson紧化和Gelfand-Kapranov-Zelevinski-Alexeev-Olsson关于次多面体的理论，同时也提出了一种将镜像作为正则描述环的Proj的综合构造。 该方案包括在二维空间中实现这一点的详细方案，以及所有维度的簇非正式地：如果你生活在一个卡-丘变量上，那么在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不同领域中起着基础性的作用，这些内禀量也应该起着类似的基础性作用。更广泛地说，弦理论模型表明我们生活在卡-丘上，因此这个提议表明，在我们的世界中存在着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.