Affine Manifolds and Mirror Symmetry

M.格罗斯计划研究卡拉比-尧曼尼褶皱的镜像对称几何。这将从M.格罗斯桑德B。西伯特与Calabi-Yau流形的某些大的复结构极限退化相联系，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。反之，给定一个具有奇点的仿射流形，则可以沿着log格式的结构构造这种退化的退化纤维。格罗斯首先计划通过证明这些对数方案可以被平滑来完成具有奇异性的仿射流形和卡-丘变量的大的复杂结构极限退化之间的对应。此外，Grossplans计算这些smoothings的不变量的仿射流形上的结构。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计算。最终的目标是将这些结果与Gromov-Witten不变量的计算结果进行比较，从而最终解释镜像对称性。格罗斯处于弦理论和几何学的交叉点。弦理论用一小圈穿过时空的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。既然时空看起来是四维的，那么我们可以预期其中的六个维度是一个非常小的“卷曲”的几何对象。这些几何对象被称为卡-丘流形。在20世纪90年代早期，弦理论家提出了完全不同的卡-丘流形之间的一种显著联系：某些在一个卡-丘流形上极难执行的计算可以通过在不同的卡-丘流形上执行完全不同的、容易得多的计算来完成。这一发现被称为镜像对称。从那时起，许多几何学家一直试图理解这一奇迹般的观察背后的数学原理。M的工作。格罗斯肖普斯对镜像对称现象给出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.