Intersection Theory for Non Intersectional Cycles

在这个项目中，研究者研究了两个循环的交集，它们的维度加起来不等于周围空间的维度。在以前的交叉理论研究中，这种情况被忽略了，因为通常这些循环不满足（因此我们称之为非交叉循环）。周期空间对我们来说是一个很大的谜题，而这个非相交的情况是这个谜题中仍然缺失的一部分。为了得到循环空间的完整图像，我们还应该包括非相交的循环。研究者研究的第一个案例是链接案例，其中循环的维度加起来比周围空间的维度少1。基本的方法是，这样的交集理论不仅应该包括相遇的循环，也应该包括那些不相遇的循环。为了完成这一任务，研究者借用了Arakelov几何中的工具-阿基米德高度配对（或者更准确地说，是由Gillet和Soule开发的算术相交理论），该理论仅定义为不满足的连接循环对。在这个方向上，研究者取得了显著的进展：(1)他得到了阿基米德高度对渐近的主导项的公式。(2)研究Mazur的发病率结构，他在Chow品种上构造了发病率因子。(3)在以上两个结果的基础上，他给出了克莱门斯猜想的证明：一般五次三折只允许每一次光滑有理曲线的有限多条。我们的计划是进一步理解这个包括一般非相交周期的相交理论。该项目集中在(1)研究关联除数；(2)研究关联等价与Abel-Jacobi等价之间的关系；(3)应用于研究Chow群与Chow品种之间的关系；(4)应用于构建Chow群上的Beilinson-Bloch滤波。数学中最基本的问题之一是代数方程的求解。一旦人们意识到人们不可能总是明确地写下方程的解，范式就变成了检验不同类型问题的模式，比如：一个解是否存在，如果存在，有多少个解，解集是否有附加结构？这些是代数几何中的基本问题。为了回答这些问题，数学家们开发了各种各样的技术，其中之一是交点理论——研究两个或多个方程组解集的交点。在这个项目中，研究者计划开发一种新的交叉理论方法。这个项目的意义在于研究当前交叉理论技术研究较少或完全未触及的材料。

In this project the investigator studies the intersection of two cycles whose dimensions do not add up to the dimension of the ambient space. In previous studies of intersection theory this is the case that has been ignored, since in general these cycles do not meet (thus we call them non-intersectional cycles). The space of cycles is a big puzzle to us, and this non-intersectional case is a piece that is still missing from the puzzle. So in order to have a complete picture of space of cycles, one should also include non-intersectional cycles. The first case studied by the investigator was the linking case where the dimensions of cycles add up to the number that is one less than the dimension of the ambient space. The fundamental approach is that such an intersection theory should not only include the cycles that meet, but also those that do not meet. To accomplish this the investigator borrows a tool-Archimedean height pairing from Arakelov geometry (or to be precise, the Arithmetic intersection theory developed by Gillet and Soule), which is only defined for pairs of linking cycles that do not meet. In this direction the investigator has made significant progress: (1) He obtained formulas for the leading term of the asymptotics of Archimedean height pairing. (2) Investigating Mazur's incidence structure, he constructed an incidence divisor on the Chow variety. (3) Based on above two results, he gave a proof of Clemens' conjecture: generic quintic three folds admit only finitely many smooth rational curves of each degree. The plan is to further the understanding of this intersection theory that includes general non-intersectional cycles. The project is concentrated in (1) the study of incidence divisors, (2) the relation between the incidence equivalence and the Abel-Jacobi equivalence, (3) the application to a study of the relation between the Chow group and the Chow variety, (4) the application to a construction of Beilinson-Bloch filtration on the Chow group. One of the most fundamental problems in mathematics is the solving of algebraic equations. Once people it was realized that one could not always explicitly write down the solutions of equations, the paradigm changed into the mode of examining different types of questions such as: does a solution exist, if so, how many solutions are there, do the solution sets have additional structure? These are the fundamental questions in algebraic geometry. In order to answer them, mathematicians have developed varied techniques, one of which-intersection theory--studies he intersection of solution sets of two or more systems of equations. In this project, the investigator plans to develop a new method in intersection theory. The significance of this project is to investigate material that is less studied or completely untouched by the current techniques of intersection theory.