Asian Journal of Mathematics

Volume 22 (2018)

Number 3

Special issue in honor of Ngaiming Mok (2 of 3)

Guest Editors: Jun-Muk Hwang, Korea Institute for Advanced Study; Yum-Tong Siu, Harvard University; Wing-Keung To, National University of Singapore; Stephen S.-T. Yau, Tsinghua University; Sai-Kee Yeung, Purdue University

On the second main theorem of Nevanlinna theory for singular divisors with $(k, \ell)$-conditions

Pages: 507 – 522



Qingchun Ji (School of Mathematical Sciences, Shanghai Center for Mathematical Sciences, Fudan University, Shanghai, China)

Guangsheng Yu (Department of Mathematics, University of Shanghai for Science and Technology, Shanghai, China)


In this paper, we introduce the $(k, \ell)$-condition on divisors by using germ decompositions and a new ramification current as the curvature current of a singular metric. Then we prove Second Main Theorem type results of Nevanlinna theory for divisors satisfying our $(k, \ell)$-condition with a new ramification term which produces an extra Characteristic Function term of a meromorphic map defined by Jacobian minors. Our Main theorem recovers Lang’s result when $\ell = 1$, and covers the general position case when $k = 1$.


ramification current, $(k, \ell)$-condition, Second Main Theorem, differentiably non-degenerate, negative curvature method

2010 Mathematics Subject Classification

32A22, 32B10, 32H30, 32J25

The authors were partially supported by NSFC 11671090 and 11322103.

Received 12 August 2016

Accepted 21 June 2017

Published 8 August 2018