Abelian branched covers and symplectic geography problem

Abstract

In this thesis, we attempt to study the geography problem of a certain class of symplectic four-manifolds. We show how branched covering techniques in algebraic geometry can be used to achieve this goal. We study line arrangements and use them to explore the geography problem further. We prove that supersolvable line arrangements can be used to construct symplectic four-manifolds with positive signatures. We give an analytic proof for the existence of a symplectic structure on algebro-geometric branched covers. We study a certain class of four-manifolds that possess the ∞ 2 - property. We finally use a special line arrangement called the Wiman arrangement to improve the asymptotic formula related to the geography problem.