Meromorphic mappings of a complete connected Kähler manifold into a projective space sharing hyperplanes

ABSTRACT

Let M be a complete Kähler manifold, whose universal covering is biholomorphic to a ball ${\mathbb{B}}^m\left(R_0\right)$ in ${\mathbb{C}}^m$ ($0<R_0\le +\mathrm{\infty }$). In this article, we will show that if three meromorphic mappings $f^1,f^2,f^3$ of M into ${\mathbb{P}}^n\left(\mathbb{C}\right)(n\ge 2)$ satisfying the condition $\left(C_{\rho }\right)$ and sharing $q(q>2n+1+\alpha +\rho K)$ hyperplanes in general position regardless of multiplicity with certain positive constants K and $\alpha <1$ (explicitly estimated), then $f^1=f^2$ or $f^2=f^3$ or $f^3=f^1$. Moreover, if the above three mappings share the hyperplanes with mutiplicity counted to level n + 1 then $f^1=f^2=f^3.$ Our results generalize the finiteness and uniqueness theorems for meromorphic mappings of ${\mathbb{C}}^m$ into ${\mathbb{P}}^n\left(\mathbb{C}\right)$ sharing 2n + 2 hyperplanes in general position with truncated multiplicity.

COMMUNICATED BY:

• S. Ivashkovich

KEYWORDS:

• Finiteness theorem
• Kähler manifold
• non-integrated defect relation

AMS SUBJECT CLASSIFICATIONS:

• Primary: 32H30
• 32A22
• Secondary: 30D35

Acknowledgments

This work was done during a stay of the author at Vietnam Institute for Advanced Study in Mathematics (VIASM). He would like to thank the institute for the support.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.04-2018.01.