Demailly showed that the Hodge conjecture is equivalent to the statement that any (p, p)-dimensional closed current with rational cohomology class can be approximated by linear combinations of integration currents. Moreover, he showed that the Hodge conjecture follows from the statement that all strongly positive closed currents with rational cohomology class can be approximated by positive linear combinations of integration currents [Dem82]. In this article, we find a current which does not verify the latter statement on a smooth projective variety for which the Hodge conjecture is known to hold. To construct this current, we extend the framework of 'tropical currents' introduced in [Bab14] from tori to toric varieties. We discuss extremality properties of tropical currents, and show that the cohomology class of a tropical current is the recession of its underlying tropical variety. The counterexample is obtained from a tropical surface in R 4 whose intersection form does not have the right signature in terms of the Hodge index theorem.There are corresponding decompositions according to the bidegree and bidimensionMost operations on smooth differential forms extend by duality to currents. For instance, the exterior derivative of a k-dimensional current T is the (k − 1)-dimensional current dT defined byThe current T is closed if its exterior derivative vanishes, and T is real if it is invariant under the complex conjugation. When T is closed, it defines a cohomology class of X, denoted {T}.The space of smooth differential forms of bidegree (p, p) contains the cone of positive differential forms. By definition, a smooth differential (p, p)-form ϕ is positive ifDually, a current T of bidimension (p, p) is strongly positive if T, ϕ ≥ 0 for every positive differential (p, p)-form ϕ on X.Integrating along complex analytic subsets of X provides an important class of strongly positive currents on X. If Z is a p-dimensional complex analytic subset of X, then the integration current [Z] is the (p, p)-dimensional current defined by integrating over the smooth locusSuppose from now on that X is an n-dimensional smooth projective algebraic variety over the complex numbers, and let p and q be nonnegative integers with p + q = n. Let us consider the following statements:(HC) The Hodge conjecture: The intersectionconsists of classes of p-dimensional algebraic cycles with rational coefficients.(HC ) The Hodge conjecture for currents: If T is a (p, p)-dimensional real closed current on X with cohomology classthen T is a weak limit of the formwhere λ ij are real numbers and Z ij are p-dimensional subvarieties of X.TROPICAL CURRENTS, EXTREMALITY, AND APPROXIMATION 3 (HC + ) The Hodge conjecture for strongly positive currents: If T is a (p, p)-dimensional strongly positive closed current on X with cohomology classthen T is a weak limit of the formwhere λ ij are positive real numbers and Z ij are p-dimensional subvarieties of X.Demailly proved in [Dem82, Théorème 1.10] that, for any smooth projective variety and q as above, HC + =⇒ H...

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.

To a tropical p-cycle V T in R n , we naturally associate a normal closed and (p, p)-dimensional current on (C * ) n denoted by T p n (V T ). Such a "tropical current" T p n (V T ) will not be an integration current along any analytic set, since its support has the form Log −1 (V T ) ⊂ (C * ) n , where Log is the coordinatewise valuation with log(|.|). We remark that tropical currents can be used to deduce an intersection theory for effective tropical cycles. Furthermore, we provide sufficient (local) conditions on tropical p-cycles such that their associated tropical currents are "strongly extremal" in D ′ p,p ((C * ) n ). In particular, if these conditions hold for the effective cycles, then the associated currents are extremal in the cone of strongly positive closed currents of bidimension (p, p) on (C * ) n . Finally, we explain certain relations between approximation problems of tropical cycles by amoebas of algebraic cycles and approximations of the associated currents by positive multiples of integration currents along analytic cycles.

We analyse the dynamics of the pullback of the map $$z \longmapsto z^m$$ z ⟼ z m on the complex tori and toric varieties. We will observe that tropical objects naturally appear in the limit, and review several theorems in tropical geometry.

We analyse the dynamics of the pullback of the map z −→ z m on the complex tori and toric varieties. We will observe that tropical objects naturally appear in the limit, and review several theorems in tropical geometry.