[Paper Review] Cohomology Jumping Loci and Relative Malcev Completion

Anthony Narkawicz|ArXiv.org|Apr 25, 2008

Algebraic structures and combinatorial models39 references3 citations

TL;DR

This paper introduces the relative Malcev completion of the fundamental group of a hyperplane arrangement complement, unifying unipotent completion and cohomology with local systems. It establishes that the relative Malcev completion is controlled by the cohomology groups $ H^1(X, \mathbb{V}_\alpha) $ and $ H^2(X, \mathbb{V}_\alpha) $, and proves that the associated spectral sequence degenerates for general representations, implying the completion is determined by the cohomology of local systems.

ABSTRACT

Two standard invariants used to study the fundamental group G of the complement X of a hyperplane arrangement are the Malcev completion of G and the cohomology groups of X with coefficients in rank one local systems. In this paper, 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. This is a prosolvable group that is tightly controlled by the cohomology groups of X with coefficients in rank one local systems. The prounipotent radical U_p of the relative completion S_p corresponds to a pronilpotent Lie algebra u_p. We provide an example of a hyperplane complement X for which this algebra is not quadratically presented. In addition, we show that if X is a hyperplane complement and Y is a subtorus of the character torus, then S_p is combinatorially determined for general p in Y. Finally, we show that the relative completion S_p is generally constant over subvarieties of the character torus.

Motivation & Objective

• To unify two standard invariants of hyperplane arrangement complements: the Malcev completion of the fundamental group and cohomology with local systems.
• To define and study the relative Malcev completion of $ \pi_1(X,x_0) $ with respect to a representation $ \boldsymbol{\rho} \to (\mathbb{C}^*)^N $, generalizing the standard Malcev completion.
• To show that the relative Malcev completion is controlled by the cohomology groups $ H^1(X, \mathbb{V}_\alpha) $ and $ H^2(X, \mathbb{V}_\alpha) $, where $ \mathbb{V}_\alpha $ are rank one local systems.
• To establish a spectral sequence degeneration result for general representations, implying the completion is determined by cohomological data.
• To prove that the Eilenberg-Moore spectral sequence for the relative completion degenerates at $ E_\infty $ for generic $ \boldsymbol{\rho} $, via a Noetherian module argument on $ \mathcal{O}(Y) $-modules.

Proposed method

• Define the relative Malcev completion $ \mathcal{S}_{\boldsymbol{\rho}} $ as a proalgebraic group extension $ 1 \to \mathcal{U}_{\boldsymbol{\rho}} \to \mathcal{S}_{\boldsymbol{\rho}} \to D_{\boldsymbol{\rho}} \to 1 $, where $ D_{\boldsymbol{\rho}} $ is the Zariski closure of $ \boldsymbol{\rho}(\pi_1(X,x_0)) $ in $ (\mathbb{C}^*)^N $.
• Use rational homotopy theory to construct the pronilpotent Lie algebra $ \mathfrak{u}_{\boldsymbol{\rho}} $ from the cdga $ E^\bullet(X, \mathcal{O}_{\boldsymbol{\rho}}) = \bigoplus_{\alpha \in D_{\boldsymbol{\rho}}^\vee} E^\bullet(X, \mathbb{V}_\alpha) \otimes V_\alpha^* $.
• Establish $ D_{\boldsymbol{\rho}} $-equivariant isomorphisms: $ \prod_{\alpha} H^1(X, \mathbb{V}_\alpha)^* \otimes V_\alpha \cong H_1(\mathfrak{u}_{\boldsymbol{\rho}}) $ and a surjection $ \prod_{\alpha} H^2(X, \mathbb{V}_\alpha)^* \otimes V_\alpha \twoheadrightarrow H_2(\mathfrak{u}_{\boldsymbol{\rho}}) $.
• Construct the Eilenberg-Moore spectral sequence $ E_n(Y) $ for the reduced bar construction over $ \mathcal{O}(Y) $, with $ E_1^{-s,t}(Y) = [H^+(X, \mathcal{O}_Y)^{\otimes s}]^t \otimes_{\mathcal{O}(Y)} \mathcal{O}(D_Y) $.
• Prove that $ E_n(Y) $ is a complex of countably generated $ \mathcal{O}(Y) $-modules, and use Theorem 9.14 to lift isomorphisms from $ E_1 $ to $ E_\infty $ via Noetherianity.
• Show that for general $ \boldsymbol{\rho} \in Y $, the canonical map $ E_\infty(Y) \otimes_{\mathcal{O}(Y)} \mathbb{C}_{\boldsymbol{\rho}} \to E_\infty(\boldsymbol{\rho}) $ is an isomorphism, using density of image and cohomology base change.

Experimental results

Research questions

• RQ1 How can the Malcev completion of the fundamental group of a hyperplane arrangement complement be generalized to incorporate representations into $ (\mathbb{C}^*)^N $?
• RQ2 What is the precise relationship between the cohomology groups $ H^1(X, \mathbb{V}_\alpha) $ and $ H^2(X, \mathbb{V}_\alpha) $ and the structure of the relative Malcev completion?
• RQ3 Under what conditions does the Eilenberg-Moore spectral sequence for the relative completion degenerate at $ E_\infty $?
• RQ4 How does the relative Malcev completion behave under base change to a general point $ \boldsymbol{\rho} \in Y $, where $ Y $ is a subvariety of $ (\mathbb{C}^*)^N $?
• RQ5 To what extent is the relative Malcev completion determined by the cohomology of local systems, and when is it quadratically presented?

Key findings

• The relative Malcev completion $ \mathcal{S}_{\boldsymbol{\rho}} $ is a proalgebraic group extension $ 1 \to \mathcal{U}_{\boldsymbol{\rho}} \to \mathcal{S}_{\boldsymbol{\rho}} \to D_{\boldsymbol{\rho}} \to 1 $, universal among such extensions lifting $ \boldsymbol{\rho} $.
• The pronilpotent Lie algebra $ \mathfrak{u}_{\boldsymbol{\rho}} $ is constructed from the cdga $ E^\bullet(X, \mathcal{O}_{\boldsymbol{\rho}}) $, and its homology is controlled by $ H^1(X, \mathbb{V}_\alpha) $ and $ H^2(X, \mathbb{V}_\alpha) $ via $ D_{\boldsymbol{\rho}} $-equivariant maps.
• For general $ \boldsymbol{\rho} \in Y $, the canonical map $ E_\infty(Y) \otimes_{\mathcal{O}(Y)} \mathbb{C}_{\boldsymbol{\rho}} \to E_\infty(\boldsymbol{\rho}) $ is an isomorphism, implying spectral sequence degeneration at $ E_\infty $.
• The Eilenberg-Moore spectral sequence $ E_n(Y) $ is a complex of countably generated $ \mathcal{O}(Y) $-modules, and this property allows inductive lifting of isomorphisms from $ E_1 $ to $ E_\infty $ via Theorem 9.14.
• When $ \boldsymbol{\rho} $ is Zariski dense in $ G_Y $, the cohomology base change $ H^\bullet(X, \mathcal{O}_Y) \otimes_{\mathcal{O}(Y)} \mathbb{C}_{\boldsymbol{\rho}} \to H^\bullet(X, \mathcal{O}_{\boldsymbol{\rho}}) $ is an isomorphism, ensuring compatibility of cohomological data.
• The relative Malcev completion is not in general 1-formal, as shown by a counterexample in the paper, indicating that Massey products may not vanish modulo indeterminacy.