# Regularization of closed positive currents and intersection theory

Complex Manifolds (2017)

• Volume: 4, Issue: 1, page 120-136
• ISSN: 2300-7443

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## Abstract

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We prove the existence of a closed regularization of the integration current associated to an effective analytic cycle, with a bounded negative part. By means of the King formula, we are reduced to regularize a closed differential form with L1loc coefficients, which by extension has a test value on any positive current with the same support as the cycle. As a consequence, the restriction of a closed positive current to a closed analytic submanifold is well defined as a closed positive current. Lastly, given a closed smooth differential (qʹ, qʹ)-form on a closed analytic submanifold, we prove the existence of a closed (qʹ, qʹ)-current having a restriction equal to that differential form. After blowing up we deal with the case of a hypersurface and then the extension current is obtained as a solution of a linear differential equation of order 1.

## How to cite

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Michel Méo. "Regularization of closed positive currents and intersection theory." Complex Manifolds 4.1 (2017): 120-136. <http://eudml.org/doc/288280>.

@article{MichelMéo2017,
abstract = {We prove the existence of a closed regularization of the integration current associated to an effective analytic cycle, with a bounded negative part. By means of the King formula, we are reduced to regularize a closed differential form with L1loc coefficients, which by extension has a test value on any positive current with the same support as the cycle. As a consequence, the restriction of a closed positive current to a closed analytic submanifold is well defined as a closed positive current. Lastly, given a closed smooth differential (qʹ, qʹ)-form on a closed analytic submanifold, we prove the existence of a closed (qʹ, qʹ)-current having a restriction equal to that differential form. After blowing up we deal with the case of a hypersurface and then the extension current is obtained as a solution of a linear differential equation of order 1.},
author = {Michel Méo},
journal = {Complex Manifolds},
keywords = {Chern class; Green operator; MacPherson graph construction; Modification; Positive current; Residue current},
language = {eng},
number = {1},
pages = {120-136},
title = {Regularization of closed positive currents and intersection theory},
url = {http://eudml.org/doc/288280},
volume = {4},
year = {2017},
}

TY - JOUR
AU - Michel Méo
TI - Regularization of closed positive currents and intersection theory
JO - Complex Manifolds
PY - 2017
VL - 4
IS - 1
SP - 120
EP - 136
AB - We prove the existence of a closed regularization of the integration current associated to an effective analytic cycle, with a bounded negative part. By means of the King formula, we are reduced to regularize a closed differential form with L1loc coefficients, which by extension has a test value on any positive current with the same support as the cycle. As a consequence, the restriction of a closed positive current to a closed analytic submanifold is well defined as a closed positive current. Lastly, given a closed smooth differential (qʹ, qʹ)-form on a closed analytic submanifold, we prove the existence of a closed (qʹ, qʹ)-current having a restriction equal to that differential form. After blowing up we deal with the case of a hypersurface and then the extension current is obtained as a solution of a linear differential equation of order 1.
LA - eng
KW - Chern class; Green operator; MacPherson graph construction; Modification; Positive current; Residue current
UR - http://eudml.org/doc/288280
ER -

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