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Convexity of complements of tropical varieties, and approximations of currents

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Abstract

The goal of this note is to affirm a local version of conjecture of Nisse–Sottile [19] on higher convexity of complements of tropical varieties, while providing a family of counter-examples for the global Nisse–Sottle conjecture in any codimension and dimension higher than one. Moreover, it is shown that, surprisingly, this family also provides a family of counter-examples for the generalized Hodge conjecture for positive currents in these dimensions, and gives rise to further approximability obstruction.

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Notes

  1. Throughout this article, by Hausdorff limit, we mean Hausdorff limit with respect to compact subsets of \(\mathbb {R}^n\).

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Acknowledgements

The authors thank the referees for the fruitful comments. We thank the CRM Ennio de Giorgi for its hospitality that enabled much of this work. Additionally, we are grateful to Emanuele Delucchi, Omid Amini, Romain Dujardin, Charles Favre, and Pierre Schapira for the encouraging discussions, and their support, and we are thankful to Nessim Sibony for the communications and references.

Author information

Authors and Affiliations

  1. Hebrew University of Jerusalem, Jerusalem, Israel

    Karim Adiprasito

  2. Département de Mathématiques, Université de Fribourg, Fribourg, Switzerland

    Farhad Babaee

Authors
  1. Karim Adiprasito
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  2. Farhad Babaee
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Corresponding author

Correspondence to Karim Adiprasito.

Additional information

Communicated by Jean-Yves Welschinger.

Karim Adiprasito is supported by ERC StG 716424—CASe and ISF Grant 1050/16. Farhad Babaee is supported by the SNSF:PP00P2 150552/1 Grant.

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Adiprasito, K., Babaee, F. Convexity of complements of tropical varieties, and approximations of currents. Math. Ann. 373, 237–251 (2019). https://doi.org/10.1007/s00208-018-1728-2

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