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Equidistribution in higher codimension for holomorphic endomorphisms of $ \mathbb{P}^k$


Author: Taeyong Ahn
Journal: Trans. Amer. Math. Soc. 368 (2016), 3359-3388
MSC (2010): Primary 37F10, 32H50, 32U40
DOI: https://doi.org/10.1090/tran/6539
Published electronically: May 4, 2015
MathSciNet review: 3451880
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Abstract: In this paper, we discuss the equidistribution phenomena for holomorphic endomorphisms over $ \mathbb{P}^k$ in the case of bidegree $ (p,p)$ with $ 1\leq p\leq k$, in particular, $ 1<p<k$. We prove that if $ f:\mathbb{P}^k\to \mathbb{P}^k$ is a holomorphic endomorphism of degree $ d\geq 2$ and $ T^p$ denotes the Green $ (p,p)$-current associated with $ f$, then there exists a proper invariant analytic subset $ E$ for $ f$ such that $ d^{-pn}(f^n)^*(S)$ converges to $ T^p$ exponentially fast in the current sense for every positive closed $ (p,p)$-current $ S$ of mass $ 1$ which is smooth on $ E$.


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Additional Information

Taeyong Ahn
Affiliation: Center for Geometry and its Applications, Pohang University of Science and Technology, Pohang City 790-784, The Republic of Korea
Email: triumph@postech.ac.kr

DOI: https://doi.org/10.1090/tran/6539
Keywords: Green current, equidistribution, exceptional set, super-potential
Received by editor(s): March 12, 2014
Published electronically: May 4, 2015
Additional Notes: The research of the author was supported in part by SRC-GaiA (Center for Geometry and its Applications), the Grant 2011-0030044 from The Ministry of Education, The Republic of Korea.
Article copyright: © Copyright 2015 American Mathematical Society

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