In this article, we obtain a class of tame maximal weights (Zhou weights). Using Tian functions (the function of jumping numbers with respect to the exponents of a holomorphic function or the multiples of a plurisubharmonic function) as a main tool, we establish an expression of relative types (Zhou numbers) to these tame maximal weights in integral form, which shows that the relative types satisfy tropical multiplicativity and tropical additivity. Thus, the relative types to Zhou weights are valuations (Zhou valuations) on the ring of germs of holomorphic functions. We use Tian functions and Zhou numbers to measure the singularities of plurisubharmonic functions, involving jumping numbers and multiplier ideal sheaves. Especially, the relative types to Zhou weights characterize the division relations of the ring of germs of holomorphic functions. Finally, we consider a global version of Zhou weights on domains in Cn, which is a generalization of the pluricomplex Green functions, and we obtain some properties of them, including continuity and some approximation results.

Publication:

Advances in Mathematics

http://dx.doi.org/10.1016/j.aim.2025.110364

Author:

Shijie Bao

Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China

bsjie@amss.ac.cn

Qi’an Guan

School of Mathematical Sciences, Peking University, Beijing, 100871, China

guanqian@math.pku.edu.cn

Zhitong Mi

School of Mathematics and Statistics, Beijing Jiaotong University, Beijing, 100044, China

zhitongmi@amss.ac.cn

Zheng Yuan

State Key Laboratory of Mathematical Sciences, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China

yuanzheng@amss.ac.cn