Families of Curves on Higher Dimensional Complex Projective Manifolds

9970412“高维复杂投影流形上的曲线计数”研究项目的目标是：1)明确地计算高维流形上的曲线（单维子流形）的期望数量，在期望有有限多条曲线的情况下；2)确定（刚性）曲线实际实现期望数量的时间。其中一个目的是简化Givental、Yau等人在Calabi-Yau案例中进行的三倍研究。另一种方法是研究一种特殊情况下的一般变形，在这种情况下，曲线虽然被认为是有限的，但实际上却是成簇移动的。这个项目属于复射影几何的研究领域，即研究多变量多项式方程组解的几何性质。该研究涉及量子物理学和微分方程中的基本问题。

9970412The goals of the research project 'Counting curves on higher dimensional complex projective manifolds' are1) to explicitly calculate the expected number of curves (one-complex-dimensional submanifolds) on higher dimensional manifolds in settings where one expects finitely many such curves, 2) to determine when the expected number is actually realized by (rigid) curves. One aim is to simplify recent advances by Givental, Yau, et al. carrying out 1) in the case of Calabi-Yau threefolds. Another is to study the general deformation of a special situation in which the curves, although expected to be finite, actually move in families.This project lies within the area of research called complex projective geometry, that is, the study of geometric properties of solutions of systems of polynomial equations in several variables. The research is related to fundamental questions in quantum physics and differential equations.