Degeneracy of holomorphic mappings into or avoiding Fermat type hypersurfaces

Abstract

We prove that if $f:{\mathbb{C}}^p\to {\mathbb{P}}^n(\mathbb{C})$ is a holomorphic mapping of maximal rank whose image lies in the Fermat hypersurface of degree $d>(n+1)\max \{n-p,1\}$, then its image is contained in a linear subspace of dimension at most $\left[\frac{n-1}{2}\right]$. Analog in the logarithmic case is also given. Our result strengthens a classical result of Green and provides a Nevanlinna theoretic proof for a recent result due to Etesse.

arXiv comment10 pages, improves arXiv:2306.14743, small modifications