# Pull-back of currents by meromorphic maps

• Volume: 141, Issue: 4, page 517-555
• ISSN: 0037-9484

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## Abstract

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Let $X$ and $Y$ be compact Kähler manifolds, and let $f:X\to Y$ be a dominant meromorphic map. Based upon a regularization theorem of Dinh and Sibony for DSH currents, we define a pullback operator ${f}^{♯}$ for currents of bidegrees $\left(p,p\right)$ of finite order on $Y$ (and thus foranycurrent, since $Y$ is compact). This operator has good properties as may be expected. Our definition and results are compatible to those of various previous works of Meo, Russakovskii and Shiffman, Alessandrini and Bassanelli, Dinh and Sibony, and can be readily extended to the case of meromorphic correspondences. We give an example of a meromorphic map $f$ and two nonzero positive closed currents ${T}_{1},{T}_{2}$ for which ${f}^{♯}\left({T}_{1}\right)=-{T}_{2}$. We use Siu’s decomposition to help further study on pulling back positive closed currents. Many applications on finding invariant currents are given.

## How to cite

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Trung Truong, Tuyen. "Pull-back of currents by meromorphic maps." Bulletin de la Société Mathématique de France 141.4 (2013): 517-555. <http://eudml.org/doc/272547>.

@article{TrungTruong2013,
abstract = {Let $X$ and $Y$ be compact Kähler manifolds, and let $f:X\rightarrow Y$ be a dominant meromorphic map. Based upon a regularization theorem of Dinh and Sibony for DSH currents, we define a pullback operator $f^\{\sharp \}$ for currents of bidegrees $(p,p)$ of finite order on $Y$ (and thus foranycurrent, since $Y$ is compact). This operator has good properties as may be expected. Our definition and results are compatible to those of various previous works of Meo, Russakovskii and Shiffman, Alessandrini and Bassanelli, Dinh and Sibony, and can be readily extended to the case of meromorphic correspondences. We give an example of a meromorphic map $f$ and two nonzero positive closed currents $T_1,T_2$ for which $f^\{\sharp \}(T_1)=-T_2$. We use Siu’s decomposition to help further study on pulling back positive closed currents. Many applications on finding invariant currents are given.},
author = {Trung Truong, Tuyen},
journal = {Bulletin de la Société Mathématique de France},
keywords = {currents; dominant meromorphic maps; untersection of currents; pull-back of currents},
language = {eng},
number = {4},
pages = {517-555},
publisher = {Société mathématique de France},
title = {Pull-back of currents by meromorphic maps},
url = {http://eudml.org/doc/272547},
volume = {141},
year = {2013},
}

TY - JOUR
AU - Trung Truong, Tuyen
TI - Pull-back of currents by meromorphic maps
JO - Bulletin de la Société Mathématique de France
PY - 2013
PB - Société mathématique de France
VL - 141
IS - 4
SP - 517
EP - 555
AB - Let $X$ and $Y$ be compact Kähler manifolds, and let $f:X\rightarrow Y$ be a dominant meromorphic map. Based upon a regularization theorem of Dinh and Sibony for DSH currents, we define a pullback operator $f^{\sharp }$ for currents of bidegrees $(p,p)$ of finite order on $Y$ (and thus foranycurrent, since $Y$ is compact). This operator has good properties as may be expected. Our definition and results are compatible to those of various previous works of Meo, Russakovskii and Shiffman, Alessandrini and Bassanelli, Dinh and Sibony, and can be readily extended to the case of meromorphic correspondences. We give an example of a meromorphic map $f$ and two nonzero positive closed currents $T_1,T_2$ for which $f^{\sharp }(T_1)=-T_2$. We use Siu’s decomposition to help further study on pulling back positive closed currents. Many applications on finding invariant currents are given.
LA - eng
KW - currents; dominant meromorphic maps; untersection of currents; pull-back of currents
UR - http://eudml.org/doc/272547
ER -

## References

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