Adiprasito, Karim Babaee, Farhad 2019-02-01 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. application/pdf https://folia.unifr.ch/global/documents/307591 https://folia.unifr.ch/documents/307591/files/bab_cct.pdf eng info:eu-repo/semantics/altIdentifier/doi/10.1007/s00208-018-1728-2 info:eu-repo/semantics/openAccess License undefined Mathematische Annalen. - 2019, vol. 373, no. 1, p. 237–251 info:eu-repo/classification/udc/51 Convexity of complements of tropical varieties, and approximations of currents http://purl.org/coar/resource_type/c_6501