TY - JOUR

T1 - A tropical approach to a generalized hodge conjecture for positive currents

AU - Babaee, Farhad

AU - Huh, June

N1 - Funding Information:
This research was conducted during the period in which Farhad Babaee was a Ph.D. student at the Universities of Bordeaux and Padova and a postdoc at Concordia University, École Normale Supérieure of Paris, and Université de Fribourg. Babaee’s work was partially supported by Swiss National Science Foundation grant PP00P2 150552=1. Huh’s work was supported by a Clay Research Fellowship and National Science Foundation grant DMS-1128155.
Publisher Copyright:
© 2017.

PY - 2017/10/1

Y1 - 2017/10/1

N2 - In 1982, Demailly showed that the Hodge conjecture follows from the statement that all positive closed currents with rational cohomology class can be approximated by positive linear combinations of integration currents. Moreover, in 2012, he 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. In this article, we find a current which does not verify the former 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"-recently introduced by the first author-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 ℝ4 whose intersection form does not have the right signature in terms of the Hodge index theorem.

AB - In 1982, Demailly showed that the Hodge conjecture follows from the statement that all positive closed currents with rational cohomology class can be approximated by positive linear combinations of integration currents. Moreover, in 2012, he 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. In this article, we find a current which does not verify the former 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"-recently introduced by the first author-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 ℝ4 whose intersection form does not have the right signature in terms of the Hodge index theorem.

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U2 - 10.1215/00127094-2017-0017

DO - 10.1215/00127094-2017-0017

M3 - Article

AN - SCOPUS:85030548458

SN - 0012-7094

VL - 166

SP - 2749

EP - 2813

JO - Duke Mathematical Journal

JF - Duke Mathematical Journal

IS - 14

ER -