Resonance, linear syzygies, Chen groups, and the Bernstein-Gelfand-Gelfand correspondence

If \A is a complex hyperplane arrangement, with complement X, we show that the Chen ranks of G=\pi_1(X) are equal to the graded Betti numbers of the linear strand in a minimal, free resolution of the cohomology ring A=H^*(X,\k), viewed as a module over the exterior algebra E on \A: \theta_k(G) = \dim_\k Tor^E_{k-1}(A,\k)_k, where \k is a field of characteristic 0, and k\ge 2. The Chen ranks conjecture asserts that, for k sufficiently large, \theta_k(G) =(k-1) \sum_{r\ge 1} h_r \binom{r+k-1}{k}, where h_r is the number of r-dimensional components of the projective resonance variety R^1(\A). Our earlier work on the resolution of A over E and the above equality yield a proof of the conjecture for graphic arrangements. Using results on the geometry of R^1(\A) and a localization argument, we establish the conjectured lower bound for the Chen ranks of an arbitrary arrangement \A. Finally, we show that there is a polynomial P(t) of degree equal to the dimension of R^1(\A), such that \theta_k(G) = P(k), for k sufficiently large.Comment: 21 pages; final versio

Characteristic varieties of arrangements

The k-th Fitting ideal of the Alexander invariant B of an arrangement A of n complex hyperplanes defines a characteristic subvariety, V_k(A), of the complex algebraic n-torus. In the combinatorially determined case where B decomposes as a direct sum of local Alexander invariants, we obtain a complete description of V_k(A). For any arrangement A, we show that the tangent cone at the identity of this variety coincides with R^1_k(A), one of the cohomology support loci of the Orlik-Solomon algebra. Using work of Arapura and Libgober, we conclude that all positive-dimensional components of V_k(A) are combinatorially determined, and that R^1_k(A) is the union of a subspace arrangement in C^n, thereby resolving a conjecture of Falk. We use these results to study the reflection arrangements associated to monomial groups.Comment: LaTeX2e, 20 pages. A reference to Libgober's recent work in math.AG/9801070 is added. Several points are clarified, a new example is include

Chen ranks and resonance

The Chen groups of a group $G$ are the lower central series quotients of the maximal metabelian quotient of $G$. Under certain conditions, we relate the ranks of the Chen groups to the first resonance variety of $G$, a jump locus for the cohomology of $G$. In the case where $G$ is the fundamental group of the complement of a complex hyperplane arrangement, our results positively resolve Suciu's Chen ranks conjecture. We obtain explicit formulas for the Chen ranks of a number of groups of broad interest, including pure Artin groups associated to Coxeter groups, and the group of basis-conjugating automorphisms of a finitely generated free group.Comment: final version, to appear in Advances in Mathematic

Lower central series and free resolutions of hyperplane arrangements

If $M$ is the complement of a hyperplane arrangement, and A=H^*(M,\k) is the cohomology ring of $M$ over a field of characteristic 0, then the ranks, $\phi_k$, of the lower central series quotients of $\pi_1(M)$ can be computed from the Betti numbers, b_{ii}=\dim_{\k} \Tor^A_i(\k,\k)_i, of the linear strand in a (minimal) free resolution of \k over $A$. We use the Cartan-Eilenberg change of rings spectral sequence to relate these numbers to the graded Betti numbers, b'_{ij}=\dim_{\k} \Tor^E_i(A,\k)_j, of a (minimal) resolution of $A$ over the exterior algebra $E$. From this analysis, we recover a formula of Falk for $\phi_3$, and obtain a new formula for $\phi_4$. The exact sequence of low degree terms in the spectral sequence allows us to answer a question of Falk on graphic arrangements, and also shows that for these arrangements, the algebra $A$ is Koszul iff the arrangement is supersolvable. We also give combinatorial lower bounds on the Betti numbers, $b'_{i,i+1}$, of the linear strand of the free resolution of $A$ over $E$; if the lower bound is attained for $i = 2$, then it is attained for all $i \ge 2$. For such arrangements, we compute the entire linear strand of the resolution, and we prove that all components of the first resonance variety of $A$ are local. For graphic arrangements (which do not attain the lower bound, unless they have no braid sub-arrangements), we show that $b'_{i,i+1}$ is determined by the number of triangles and $K_4$ subgraphs in the graph.Comment: 25 pages, to appear in Trans. Amer. Math. So