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Sigma-invariants and tropical varieties

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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.

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Acknowledgements

We wish to thank the referee for a careful reading of the manuscript and for many valuable comments and suggestions that helped us to improve both the substance and the exposition of the paper.

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    Alexander I. Suciu

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Suciu, A.I. Sigma-invariants and tropical varieties. Math. Ann. 380, 1427–1463 (2021). https://doi.org/10.1007/s00208-021-02172-z

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