Abstract

Abstract We study some fundamental properties of real rectifiable currents and give a generalization of King’s theorem to characterize currents defined by positive real holomorphic chains. Our main tool is Siu’s semi-continuity theorem and our proof largely simplifies King’s proof. A consequence of this result is a sufficient condition for the Hodge conjecture.

Highlights

  • Since the publication of the foundational paper “Normal and integral currents" [7] by Federer and Fleming, geometric measure theory becomes an important tool in many areas of mathematics [5]

  • We study some fundamental properties of real recti able currents and give a generalization of King’s theorem to characterize currents de ned by positive real holomorphic chains

  • The rst major progress was made by King in his marvelous paper [14] where he proved that holomorphic chains with positive integral coe cients are those d-closed recti able positive currents

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Summary

Introduction

Since the publication of the foundational paper “Normal and integral currents" [7] by Federer and Fleming, geometric measure theory becomes an important tool in many areas of mathematics [5]. King’s result [13] showing that d-closed recti able currents of type (k, k) with ( k + )-Hausdor measure 0 support are integral holomorphic chains. They conjectured that the condition on support is not necessary. Since the last condition in our main theorem that N is H k-locally nite is automatically satis ed by positive integral currents, our result generalizes King’s result and our proof largely simpli es King’s proof This simpli cation is possible because of our use of Siu’s famous semicontinuity theorem [19].

Real recti able currents
TxW for
Wi and
We have m
From the equality
Suppose that U
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