Complex Tropical Currents, Extremality, and Approximations

To a tropical $p$-cycle $V_{\mathbb{T}}$ in $\mathbb{R}^n$, we naturally associate a normal closed and $(p,p)$-dimensional current on $(\mathbb{C}^*)^n$ denoted by $\mathscr{T}_n^p(V_{\mathbb{T}})$. Such a "tropical current" $\mathscr{T}_n^p(V_{\mathbb{T}})$ will not be an integration current along any analytic set, since its support has the form ${\rm Log\,}^{-1}(V_{\mathbb{T}})\subset (\mathbb{C}^*)^n$, where ${\rm Log\,}$ is the coordinate-wise 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 $\mathcal{D}'_{p,p}((\mathbb{C}^*)^n)$. 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 $(\mathbb{C}^*)^n$. 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.