Degeneracy of holomorphic mappings into or avoiding Fermat type hypersurfaces

TL;DR

This paper proves that holomorphic maps of maximal rank into certain Fermat hypersurfaces are highly degenerate, with their images contained in low-dimensional linear subspaces, extending classical results with new Nevanlinna theoretic methods.

Contribution

It establishes a new degeneracy result for holomorphic maps into Fermat hypersurfaces, strengthening Green's classical theorem and providing a Nevanlinna theoretic proof for recent findings.

Findings

Holomorphic maps into Fermat hypersurfaces are contained in low-dimensional linear subspaces.

The degree condition $d > (n+1) imes ext{max}ig\{n-p,1ig\}$ is critical for degeneracy.

The result applies to both algebraic and logarithmic cases.

Abstract

We prove that if $f\colon\mathbb{C}^p\rightarrow\mathbb{P}^n(\mathbb{C})$ 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 $\bigg[\dfrac{n-1}{2}\bigg]$. 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.