Intersection Theory for Non Intersectional Cycles

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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.