Twisted homology jump loci, twisted Alexander polynomials, and $\Sigma$-invariants

Abstract

We introduce the twisted homology jump loci of a space $X$: the jump loci for homology with coefficients in rank-one local systems, twisted by a fixed finite-dimensional representation $\sigma$ of $\pi_1(X)$. These loci refine the classical characteristic varieties, and their defining equations in degree one are the twisted Alexander polynomials of knot theory. Our main theorem is that their tropicalizations bound from above the Bieri--Neumann--Strebel--Renz (BNSR) $\Sigma$-invariants of $X$. Twisting gains real ground. The resulting bound is strictly stronger than the untwisted tropical bounds obtained from the usual characteristic varieties: for a one-relator group whose $\Sigma^1$ was computed by Brown, the untwisted bound excludes only two directions in $H^1(G;\mathbb{R})$, whereas the twisted bound determines $\Sigma^1(G)$ exactly. For a compact orientable $3$-manifold $M$ with toroidal or empty boundary, the twisted bound is sharp: the union of the twisted tropical varieties over all finite-image integral representations of $\pi_1(M)$ computes $\Sigma^1(\pi_1(M))$, and hence recovers the fibered faces of the Thurston norm ball. Sharpness genuinely requires twisting: a non-fibered class enters the tropical variety through the vanishing of a twisted Alexander polynomial along it, and the untwisted polynomial need not vanish. For a compact K\"{a}hler manifold $X$, we prove that the first twisted Alexander polynomial $\Delta^{\sigma}(X)$ is either $0$ or $1$, for every representation $\sigma$ over every field, and that $\Sigma^1(\pi_1(X))$ is controlled by the hyperbolic orbifold fibrations of $X$ for every $\sigma$. The obstruction to K\"{a}hlerianity that comes out of this is strictly finer than its untwisted counterpart: we exhibit groups $G$ with $\Delta(G)=1$ but $\Delta^{\sigma}(G)\ne 1$, which the twisted test excludes from being K\"{a}hler and the classical one does not.