  We use stratified Morse theory to construct a complex to compute the
cohomology of the complement of a hyperplane arrangement with coefficients in a
complex rank one local system. The linearization of this complex is shown to be
the Orlik-Solomon algebra with the connection operator. Using this result, we
establish the relationship between the cohomology support loci of the
complement and the resonance varieties of the Orlik-Solomon algebra for any
arrangement, and show that the latter are unions of subspace arrangements in
general, resolving a conjecture of Falk. We also obtain lower bounds for the
local system Betti numbers in terms of those of the Orlik-Solomon algebra,
recovering a result of Libgober and Yuzvinsky. For certain local systems, our
results provide new combinatorial upper bounds on the local system Betti
numbers. These upper bounds enable us to prove that in non-resonant systems the
cohomology is concentrated in the top dimension, without using resolution of
singularities.
