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arXiv · 1502.00299

A tropical approach to a generalized Hodge conjecture for positive currents

Farhad Babaee·June Huh
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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.

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Farhad Babaee, June Huh. 2015-02-01. A tropical approach to a generalized Hodge conjecture for positive currents. https://doi.org/10.1215/00127094-2017-0017

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