Hypperplane Arrangements In Projective Spaces: The Interplay Between Their Combinatorics And The Topology Of Complements

The interplay between the topological and combinatorial invariants of the com-plements of hyperplane arrangements has been a very interesting and major theme in the study of line arrangement. A natural question is to what extend the combi-natorics of arrangements determine topology of their complements.This dissertation is mainly devoted to studying the following two topics:(1) what kind of properties of the combinatorics of hyperplane arrangements should have so that the diffeomorphic structures of their complements are determined by their combinatorics; (2) to what extend the combinatorics of a fiber-type projective line arrangement determines its topological invariants.On the first topic, I first studied the arrangements of hyperplanes in CP4, and found a large class of hyperplane arrangements, called nice arrangements, the diffeomorphic types of whose complements depend only on their combinatorics. More generally, I found classes of hyperplane arrangements in CP', called nice point arrangements, diffeomorphic types of whose complements depend only on their combinatorics too.On the second topic, torwards to show that the diffeomorphic types of com-plements of fiber-type line arrangements are rigid, I proved that their differen-tial structures are determined by the braid monodromies of their associated sub-arrangements. I also obtained a characterization of the Alexander modules of the complements of fiber-type projective line arrangements.