Affine Manifolds and Mirror Symmetry

格罗斯计划研究卡拉比-姚流形的镜像对称几何。这将从M. Grossand B. Siebert引入的strominger - you - zaslow猜想的代数-几何版本的角度来完成。结合某些大型复杂结构有限代的Calabi-Yau流形，可以定义一个对偶交复，它是具有奇点的仿射流形。相反，给定一个具有奇点的仿射流形，则可以建立这种退化的简并纤维，以及对数格式的结构。Gross首先计划通过证明这些对数格式可以平滑来完成具有奇点的仿射流形与Calabi-Yau变种的大型复杂结构极限退化之间的对应关系。此外，gross计划根据仿射流形上的结构计算这些光滑的不变量。人们期望霍奇数，更重要的是，霍奇结构的变化，可以直接从仿射流形的计算中计算出来。最终的目标是将这些结果与镜面的Gromov-Witten不变量的计算结果进行比较，从而最终为镜面对称性提供一个解释。格罗斯先生提出的工作是弦理论和几何学的交叉。弦理论用一个在时空中运动的小环弦取代了传统的点粒子概念。为了使弦理论与量子力学兼容，时空必须是十维的。由于时空看起来是四维的，人们预计其中的六个维度是一个非常小的“卷曲”的几何物体。这些几何物体被称为Calabi-Yau流形。在20世纪90年代早期，弦理论家提出了完全不同的Calabi-Yau流形之间的显著联系：在一个Calabi-Yau流形上执行极其困难的某些计算可以通过在不同的Calabi-Yau流形上执行完全不同的、更容易的计算来完成。这一发现被称为镜像对称。从那时起，许多几何学家一直在试图理解这一神奇观察背后的数学原理。格罗斯先生的工作旨在对镜像对称现象提供数学上的见解和解释。

M. Gross plans to study the geometry of mirror symmetry for Calabi-Yaumanifolds. This will be done from the perspective of an algebro-geometricversion of the Strominger-Yau-Zaslow conjecture introduced by M. Grossand B. Siebert. Associated to certain sorts of large complex structure limitdegenerations of Calabi-Yau manifolds one can define a dual intersectioncomplex, which is an affine manifold with singularities. Conversely,given an affine manifold with singularities, it is possible to buildthe degenerate fibre of such a degeneration, along with the structureof a log scheme. Gross first plans to completethe correspondence between affine manifolds with singularitiesand large complex structure limit degenerations of Calabi-Yau varietiesby showing that these log schemes can be smoothed. Furthermore, Grossplans to compute invariants of these smoothings in terms of structureson the affine manifolds. One expects Hodge numbers, and more importantly,variations of Hodge structure, can be calculated directly from computationson the affine manifold. The ultimate goal will be to compare these results with calculations of Gromov-Witten invariants for the mirror, therebyeventually providing an explanation for mirror symmetry.The work proposed by M. Gross lies at the intersection of string theoryand geometry. String theory replaces the traditional notion of the point particle with a small loop of string, moving through space-time.To make string theory compatible with quantum mechanics, space-time mustbe ten-dimensional. Since space-time appears four-dimensional, one expects six of these dimensions to be a very small `curled up' geometric object. These geometric objects are called Calabi-Yau manifolds. In the early1990s, string theorists proposed a remarkable association betweencompletely different Calabi-Yau manifolds: certain calculations extremelydifficult to perform on one Calabi-Yau manifold could be completedby performing completely different, and much easier, calculations ona different Calabi-Yau manifold. This discovery was known as mirror symmetry.Since this time, many geometers have been trying to understand the mathematics behind this miraculous observation. The work of M. Grosshopes to give mathematical insight and explanation for the phenomenonof mirror symmetry.