Convexity of complements of tropical varieties, and approximations of currents
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.
Mathematics Subject Classification14T05 14C30 32U40
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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