Sigma-invariants and tropical varieties

Abstract

The Bieri-Neumann-Strebel-Renz invariants ${\Sigma }^q(X,\mathbb{Z})\subset H^1(X,\mathbb{R})$ of a connected, finite-type CW-complex $X$ are the vanishing loci for Novikov-Sikorav homology in degrees up to $q$, while the characteristic varieties ${\mathcal{V}}^q(X)\subset H^1(X,{\mathbb{C}}^{\times })$ are the nonvanishing loci for homology with coefficients in rank 1 local systems in degree $q$. We show that each BNSR invariant ${\Sigma }^q(X,\mathbb{Z})$ is contained in the complement of the tropical variety associated to the algebraic variety ${\mathcal{V}}^{\le q}(X)$, and provide applications to several classes of groups and spaces.

30 pages; accepted for publication in Mathematische Annalen

References in corpus (4)

• Toric complexes and Artin kernels
• The Bieri-Neumann-Strebel invariants via Newton polytopes
• Virtual algebraic fibrations of Kähler groups
• Cohomology jump loci of 3-manifolds