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The characteristic varieties $V_d^i(X)$ of a (sufficiently nice) space $X$ are the cohomology jumping loci for 1-dimensional (complex) local systems on $X$. Assume that $H_1(X;\mathbb{Z}) \cong \mathbb{Z}^n$ for some $n > 0$. Then

$$V^i_d(X) = \{\ \rho\in \text{Hom}(\pi_1(X), \mathbb{C}^*)\ \ |\ \ \text{dim}\ H^i(X; \mathbb{C}_{\rho}) \geq d \ \}$$

where $\mathbb{C}_{\rho}$ is the 1-dimensional complex local system on $X$ associated to the character $\rho$. These loci are Zariski closed in the algebraic torus $\text{Hom}(\pi_1(X), \mathbb{C}^*) = (\mathbb{C}^*)^n$.

This question is concerned with the characteristic varieties $V^1_d$ of the complement of the braid arrangement $X_k\subset \mathbb{C}^k$.

General results of Arapura imply that $V_1^1(X_k)$ is a union of subtori of $(\mathbb{C}^*)^n$, some of which may be translated away from the identity $\bf{1}\in$ $(\mathbb{C}^*)^n$. In the late 90s, Cohen--Suciu showed that the components of $V_1^1(X_k)$ that contain $\bf{1}$ are two-dimensional and gave an explicit description of them. In about 2009, Settepanella found the remaining components of $V_1^1(X_k)$.

$\bf{Question}$: Is anything known about the structure of the characteristic varieties $V_d^1(X_k)$ with $d\geq 2$? In other words, for fixed $k$ and $d$, is anything known about the set of 1-dimensional local systems on $X_k$ whose cohomology is at least d-dimensional?

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Let $T$ be an irreducible component of $V^1_d(X_k)$ with $d\ge 2$. If $T$ contains $\mathbb{1}$, then $T=\{\mathbb{1}\}$. It is probably the case that all components of $V^1_d(X_k)$ pass through the identity, but I don't think that has been established, except for small values of $k$.

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