L~2-Type Invariants And Cohomology Jump Loci For Smooth Complex Quasi-Projective Varieties

Let X be a smooth complex quasi-projective variety with a fixed epimorphsim v:π1(X)(?)Z.Using v we can define a series of cover spaces of X,these cover spaces are called approximation spaces.A classical idea is to study X by using these approximation spaces.In this paper,we consider the asymptotic behaviour of invariants such as Betti numbers with all possible field coefficients and the order of the torsion subgroup of singular homology associated to v,known as the L~2-type invariants.We give relations between these L~2-type invariants and prove a L~2 version universal coefficient theorem.We can also define Alexander and Aomoto invariants associated to v.We give relations between these invariants and L~2-type invariants.For L~2-type invariants of orbifold groups,we use Fox calculus to get a concrete formula.For degree 1,we give concrete formulas to compute these limits by geometric informations of X when v is orbifold effective.The proof relies on the theory of cohomology jump loci.We give a detailed study about the degree 1 cohomology jump loci of X with arbitrary algebraically closed field coefficients.In particular,we extend part of Arapura’s result for degree 1 cohomology jump loci of X with zero characteristic field coefficients to the one with positive characteristic field coefficients.When X is a hyperplane arrangement complement,combinatoric upper bounds are given for the number of parallel positive dimensional components of degree 1 cohomology jump loci with complex coefficients.As an application,we give a positive answer to a question posed by Denham and Suciu:for any prime number p≥2,there exists a central hyperplane arrangement such that its Milnor fiber has non-trivial p-torsion in homology and p does not divide the number of hyperplanes in the arrangement.A key step of our proof is the L~2 version universal coefficient theorem we give before.In the last,we study a related problem of hyperplane arrangement:Given top degree Betti number of hyperplane arrangement complement,classify the hyperplane arrangement.This problem is first researched by S.Papadima and A.Dimca,they totally classifed hyperplane arrangement with top degree Betti number being 1 or 2.We use deleted-restriction method to extend their results to the case when top degree Betti number being 3,4 or 5.