The cohomology groups of finite cyclic covers of complexified real arrangement complements

Technical Summary: The Cohomology Groups of Finite Cyclic Covers of Complexified Real Arrangement Complements

Problem Statement
The central problem addressed in this paper is the combinatorial determination of the (co)homology groups of finite cyclic covers of hyperplane arrangement complements. While classical results by Orlik and Solomon established that the cohomology ring and homology groups of the complement itself are torsion-free and determined entirely by the intersection lattice, the situation for finite covers (including the Milnor fiber) is more complex. It is known that the (co)homology of such covers can contain torsion, and it remains an open question whether this torsion is combinatorially determined. Specifically, the paper investigates whether the torsion in the (co)homology of the Milnor fiber and general finite cyclic covers is determined by the intersection lattice, extending the Cohen-Dimca-Orlik (CDO) framework which provides combinatorial criteria for local system cohomology.

Methodology
The author employs the chamber cochain complex constructed by Yoshinaga to compute the cohomology of rank-one local systems on arrangement complements. The methodology proceeds through the following steps:

1. Local System and Covering Construction: The paper considers a finite cyclic cover $U^\phi$ of the complement $U$ of a complexified real arrangement, induced by an epimorphism $\phi: \pi_1(U) \to \mathbb{Z}_N$. This cover corresponds to a rank-one local system $L$ over the ring $R = \mathbb{Z}[\mathbb{Z}_N] \cong \mathbb{Z}[t]/(t^N-1)$.
2. Chamber Cochain Complex: Using a generic flag $F$ near the hyperplane at infinity $H_\infty$, the author utilizes the chamber cochain complex $(R[\text{ch}^\bullet(A)], d'_L)$. The cohomology of this complex is isomorphic to the cohomology of the local system $H^*(U, L)$.
3. Matrix Analysis: The boundary map $d'_L$ is analyzed via its matrix representation. By ordering chambers based on the dimension of their associated subspaces at infinity, the author constructs a matrix where the diagonal blocks correspond to terms of the form $1 - q_X^{-1}$, where $q_X$ is determined by the monodromy around the intersection $X$.
4. Rank Computation: The rank of the boundary map over an arbitrary field $K$ is estimated by replacing the variable $t$ with the permutation matrix $T_N$ (representing the generator of $\mathbb{Z}_N$). The rank of $T_N^k - I_N$ is determined by the greatest common divisor $\gcd(k, N)$. This allows for the derivation of explicit upper bounds for the Betti numbers.
5. Torsion-Free Condition: The paper introduces a combinatorial condition analogous to the CDO-condition, termed the CDO-condition for cyclic $N$-covers. This condition requires that for every dense edge $X \subset H_\infty$, $\gcd(\epsilon_X, N) = 1$, where $\epsilon_X$ is the sum of the exponents associated with hyperplanes containing $X$.

Key Contributions and Results

• Explicit Upper Bounds: The paper establishes explicit upper bounds for the Betti numbers $b_i(U^\phi, K)$ of finite cyclic covers over arbitrary fields $K$. For a complexified real arrangement of dimension $l$, the bounds are:

  • For $0 \le i \le l-1$: $b_i(U^\phi, K) \le \sum_{C \in \text{ch}_i(A)} \gcd(\epsilon_{X(C)}, N)$.
  • For $i = l$: $b_l(U^\phi, K) \le N \cdot (-1)^l \chi(U) + \sum_{C \in \text{uch}_l(A)} \gcd(\epsilon_{X(C)}, N)$.
    Here, $\chi(U)$ is the Euler characteristic, and the sum runs over chambers in the complex.

• Torsion-Freeness Theorem (Main Theorem): Under the assumption that the arrangement is a complexified real arrangement and the map $\phi$ satisfies the CDO-condition for cyclic $N$-covers (i.e., $\gcd(\epsilon_X, N) = 1$ for all dense edges $X \subset H_\infty$), the integral cohomology groups are proven to be torsion-free. Specifically:

  • $H^k(U^\phi, \mathbb{Z}) \cong \mathbb{Z}^{b_k}$ for $0 \le k \le l-1$.
  • $H^l(U^\phi, \mathbb{Z}) \cong \mathbb{Z}^{b_l + (N-1)(-1)^l \chi(U)}$.
    This result implies that under these conditions, the cohomology is completely determined by the intersection lattice and the covering degree $N$.

• Corollaries:

  • Milnor Fiber: For the Milnor fiber of an essential complexified real arrangement where the number of hyperplanes is a prime $p$, the cohomology groups are torsion-free and explicitly determined.
  • Transversality Condition: If there exists a hyperplane $H$ transverse to all others such that $\gcd(\epsilon_H, N) = 1$, the cohomology groups of the $N$-fold cyclic cover are torsion-free.

• Generalization of Previous Work: The results generalize recent findings by the author and Liu regarding complex line arrangements in $\mathbb{CP}^2$ to higher-dimensional complexified real arrangements.

Significance and Claims
The paper claims to provide a partial generalization of existing results on line arrangements to higher dimensions, specifically within the realm of complexified real arrangements. The primary significance lies in establishing a purely combinatorial criterion (the CDO-condition for cyclic covers) that guarantees the absence of torsion in the cohomology of finite cyclic covers. This offers a positive answer to the question of combinatorial determination for a specific, significant class of arrangements and covers, contrasting with known examples where torsion exists.

The author explicitly states that the torsion-freeness result is proven for complexified real arrangements and posits a conjecture that this result may extend to arbitrary essential complex arrangements, provided the associated local system satisfies the CDO-condition. The paper does not claim to have resolved the general case for all complex arrangements, nor does it propose new experimental applications, but rather focuses on refining the theoretical understanding of the relationship between combinatorial data and topological invariants in arrangement theory.