Uniqueness of holomorphic mappings concerning a question of gross

Abstract

A hypersurface X (resp., a pair of hypersurfaces $\{Y,Z\}$) is called a hypersurface (resp., a pair of hypersurfaces) of uniqueness for holomorphic mappings from $\mathbb{C}$ to ${\mathbb{P}}^N(\mathbb{C})$ if for two non-degenerate holomorphic mappings $f,g$ from $\mathbb{C}$ to $P^N(\mathbb{C}),$ the condition ${\nu }_f(X)={\nu }_g(X)$ (resp., ${\nu }_f(Y)={\nu }_g(Y),$ ${\nu }_f(Z)={\nu }_g(Z)$) implies $f\equiv g,$ where for a mapping φ, ${\nu }_{\phi }(V)$ denotes the pull-back of a divisor V in ${\mathbb{P}}^N(\mathbb{C})$ by φ. In this paper, we give some classes of hypersurfaces and of pairs of hypersurfaces of uniqueness for holomorphic mappings from $\mathbb{C}$ to ${\mathbb{P}}^N(\mathbb{C}).$

Keywords:

• Holomorphic mappings
• uniqueness theorems
• hypersurfaces

AMS Subject Classifications:

• 32H02
• 32H30

Acknowledgments

The authors are very grateful to the referee for carefully reading the manuscript and for the valuable suggestions.

Disclosure statement

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

Additional information

Funding

This work was supported by National Foundation for Science and Technology Development [grant number 101.02-2018.301].