Affine Manifolds and Mirror Symmetry

格罗斯计划研究Calabi-Yaumanifold的镜像对称几何。这将从M.Grosand B.Siebert提出的Strominger-Yau-Zaslow猜想的代数几何版本的角度进行。与Calabi-Yau流形的某些大型复结构极限代相联系，我们可以定义一个对偶交复形，它是一个具有奇点的仿射流形。相反，给定一个具有奇点的仿射流形，就有可能建立这种退化的简并纤维以及对数方案的结构。Gross First计划通过证明这些对数格式是可以光滑的，来完成具有奇点的仿射流形和大型复杂结构极限Calabi-Yau变元之间的对应。此外，Gross计划根据仿射流形上的结构来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上的计算中计算出来。最终目的是将这些结果与镜的Gromov-Witten不变量的计算进行比较，从而最终为镜的对称性提供解释。弦理论用一个穿越时空的小弦环取代了传统的点粒子的概念。要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，人们预计这些维度中的六个会是一个非常小的“卷曲”的几何物体。这些几何对象称为Calabi-Yau流形。在20世纪90年代初，弦理论家提出了一个完全不同的Calabi-Yau流形之间的显著联系：在一个Calabi-Yau流形上执行某些极难执行的计算，可以通过在不同的Calabi-Yau流形上执行完全不同且容易得多的计算来完成。这一发现被称为镜面对称性。从那时起，许多几何学家一直在试图理解这一奇迹观测背后的数学原理。格罗斯库斯为镜像对称现象提供了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.