2014/03/28 by Farhad Babaee, Babaee, Farhad · 1 citation
Computer Science · Mathematics · #14M25 #14T05 #32C30 #42B05 #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.1403.7456
openalex publication_date 2014/03/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
To a tropical p-cycle V_\mathbbT in ℝn, we naturally associate a normal closed and (p,p)-dimensional current on (ℂ^*)n denoted by \mathscrTnp(V_\mathbbT). Such a "tropical current" \mathscrTnp(V_\mathbbT) will not be an integration current along any analytic set, since its support has the form \rm Log -1(V_\mathbbT)⊂ (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 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.