A tropical approach to a generalized Hodge conjecture for positive currents

TL;DR

This paper uses tropical geometry to construct a counterexample to a strengthened form of the Hodge conjecture involving positive currents, challenging previous assumptions about their approximation by subvarieties.

Contribution

It introduces a novel tropical geometric method to construct a (p,p)-dimensional current that defies approximation by positive linear combinations of subvariety integration currents.

Findings

Constructed a (p,p)-dimensional current violating the approximation property.

Demonstrated limitations of positive currents approximation using tropical geometry.

Challenged existing conjectures about positive currents in complex geometry.

Abstract

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 associated to subvarieties, and asked whether any strongly positive (p,p)-dimensional closed current with rational cohomology class can be approximated by positive linear combinations of integration currents associated to subvarieties. Using tropical geometry, we construct a (p,p)-dimensional current on a smooth projective variety that does not satisfy the latter statement.