Convexity of complements of tropical varieties, and approximations of currents

Abstract

The goal of this note is to affirm a local version of conjecture of Nisse-Sottile [NS16] on higher convexity of complements of tropical varieties, while providing a family of counterexamples for the global Nisse-Sottle conjecture in any codimension and dimension higher than 1. 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.

Dynamical Tropicalisation

“…The extremality results in [5] were subsequently improved and extended to toric varieties by Huh and the author in [6], and tropical currents were used to find a non-trivial example of a positive closed current on a smooth projective toric variety which refutes a strong version of the Hodge conjecture for positive currents. Thereafter, Adiprasito and the author in [1] proposed a family of tropical currents which are counter-example to the aforementioned conjecture in any dimension and codimension greater than one. Moreover, tropical currents were also used in application to higher convexity problems and Nisse-Sottile conjecture [62], as well as finding a family of peculiar currents which cannot be regularised to obtain mollified currents with smooth boundaries; see [1,Theorem D].…”

“…Thereafter, Adiprasito and the author in [1] proposed a family of tropical currents which are counter-example to the aforementioned conjecture in any dimension and codimension greater than one. Moreover, tropical currents were also used in application to higher convexity problems and Nisse-Sottile conjecture [62], as well as finding a family of peculiar currents which cannot be regularised to obtain mollified currents with smooth boundaries; see [1,Theorem D].…”

“…where δ z is the Dirac measure at z, μ(S 1 ) is the Haar measure on S 1 , and the limit is in the weak sense of measures. Brolin's remarkable theorem [12] is a generalisation of this observation: for any monic algebraic map of f : C −→ C of degree d ≥ 2, there exists a (harmonic) measure μ f , such that for any generic point z ∈ C,…”

Dynamical Tropicalisation

“…The extremality results in [Bab14] were subsequently improved and extended to toric varieties by Huh and the author in [BH17], and tropical currents were used to find a non-trivial example of a positive closed current on a smooth projective toric variety which refutes a strong version of the Hodge conjecture for positive currents. Thereafter, Adiprasito and the author in [AB19] proposed a family of tropical currents which are counter-example to the aforementioned conjecture in any dimension and codimension greater than one. Moreover, tropical currents were also used in application to higher convexity problems and Nisse-Sottile conjecture [NS16], as well as finding a family of peculiar currents which cannot be regularised to obtain mollified currents with smooth boundaries; see [AB19,Theorem D].…”

“…Thereafter, Adiprasito and the author in [AB19] proposed a family of tropical currents which are counter-example to the aforementioned conjecture in any dimension and codimension greater than one. Moreover, tropical currents were also used in application to higher convexity problems and Nisse-Sottile conjecture [NS16], as well as finding a family of peculiar currents which cannot be regularised to obtain mollified currents with smooth boundaries; see [AB19,Theorem D]. In all the above-mentioned works though, the notion of tropicalisation of algebraic varieties was absent, which is the topic of this article.…”