Families of Curves on Higher Dimensional Complex Projective Manifolds

9970412研究项目“高维复射影流形上的曲线计数”的目标是1）明确计算高维流形上的曲线（一复维子流形）的预期数量，其中人们预期有许多这样的曲线，2）确定预期数量何时实际上由（刚性）曲线实现。 一个目的是简化Givental，Yau等人的最新进展，在Calabi-Yau三重的情况下进行1）。 另一个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，但实际上是在族中移动的。这个项目属于称为复射影几何的研究领域，即研究多变量多项式方程组的解的几何性质。 该研究涉及量子物理和微分方程的基本问题。

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.