Cohomology Jumping Loci and the Relative Malcev Completion
Abstract
Two standard invariants used to study the fundamental group of the complement X of
a hyperplane arrangement are the Malcev completion of its fundamental group G and
the cohomology groups of X with coefficients in rank one local systems. In this thesis,
we develop a tool that unifies these two approaches. This tool is the Malcev completion
S_p of G relative to a homomorphism p from G into (C^*)^N. The relative completion
S_p is a prosolvable group that generalizes the classical Malcev completion; when
p is the trivial representation, S_p is the Malcev completion of G. The group S_p
is tightly controlled by the cohomology groups H^1(X,L_{p^k}) with coefficients in
the irreducible local systems L_{p^k} associated to the representation p.The pronilpotent
Lie algebra u_p of the prounipotent radical U_p of S_p has been described by Hain.
If p is the trivial representation, then u_p is the holonomy Lie algebra, which is
well-known to be quadratically presented. In contrast, we show that when X is the
complement of the braid arrangement in complex two-space, there are infinitely many
representations p from G into (C^*)^2 for which u_p is not quadratically presented.We
show that if Y is a subtorus of the character torus T containing the trivial character,
then S_p is combinatorially determined for general p in Y. We do not know whether
S_p is always combinatorially determined. If S_p is combinatorially determined for
all characters p of G, then the characteristic varieties of the arrangement X are
combinatorially determined.When Y is an irreducible subvariety of T^N, we examine
the behavior of S_p as p varies in Y. We define an affine group scheme S_Y over Y
such that if Y = {p}, then S_Y is the relative Malcev completion S_p. For each p in
Y, there is a canonical homomorphism of affine group schemes from S_p into the affine
group scheme which is the restriction of S_Y to p. This is often an isomorphism. For
example, if there exists p in Y whose image is Zariski dense in G_m^N, then this homomorphism
is an isomorphism for general p in Y.
Type
DissertationDepartment
MathematicsSubject
MathematicsMalcev Completion
Hyperplane Arrangements
Characteristic Varieties
Orlik
Solomon Algebra
Rational Homotopy Theory
Iterated Integrals
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https://hdl.handle.net/10161/441Citation
Narkawicz, Anthony Joseph (2007). Cohomology Jumping Loci and the Relative Malcev Completion. Dissertation, Duke University. Retrieved from https://hdl.handle.net/10161/441.Collections
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