The goal of this note is to affirm a local version of conjecture of Nisse–Sottile  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.
Bibliographical noteFunding Information:
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..
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.
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