曹锡华代数论坛

报告人：丘成栋教授 （伊利诺伊大学）

时间：2010年12月13日(周一)下午15:00-16:00

地点：闵行校区数学楼102报告厅

Title: Relationship between Combinatorics and Topological Structures of Line Arrangements

Abstract:

One of the central problems in arrangement theory is to find the relationship between combinatorial structure of line arrangement in CP^2 and the topological structure of the complement of line arrangement. One direction of the research was completed by Jiang and Yau. They showed that the combinatorial structure of line arrangement in CP^2 is a topological invariance of the complement of the line arrangement. It has been a great interest for people to find out whether the combinatorial structure of line arrangement in CP^2 will determine the topological structure of the complement.

A family of line arrangements whose combinatorial structure determines the topological structure of the complement was introduced by Jiang and Yau. They defined the concept of a nice arrangement purely in terms of combinatorial structure. They proved that for two nice arrangements, if their combinatorial structures are isomorphic, then their complements are diffeomorphic. Wang and Yau continued this direction and proved that the results of JiangYau hold for a much larger family of arrangements. Rybnikov has shown that in the complex case, there is a pair of line arrangements with the same combinatorial structure, but non-isomorphic fundamental groups of the complements.

There are two interesting open problems.

(1) For a real line arrangement, does the combinatorial structure determine the fundamental group of its complement?

(2) For fiber-type (or equivalently, super-solvable) line arrangement in CP^2, does the combinatorial structure determine the topological type of the arrangement? Problem (1) above possibly will have a counterexample.

As for Problem (2) above, my recent joint work with Shengli Tan and Fei Ye shows that the central projecting monodromy determines the diffeomorphic type of the complement of fiber-type line arrangement in CP^2.