Generalized Demailly-Semple jet bundles and holomorphic mappings into complex manifolds

TL;DR

This paper extends jet-bundle techniques to study maximal rank holomorphic maps from higher-dimensional complex spaces into manifolds, proving non-existence results for certain hypersurfaces, advancing understanding of the Green-Griffiths conjecture.

Contribution

It generalizes jet-bundle methods to higher-dimensional domains and establishes non-existence of specific holomorphic maps into general hypersurfaces.

Findings

Non-existence of maximal rank holomorphic maps from  into degree  hypersurfaces for d  93.

Extension of jet-bundle techniques to -dimensional domains.

Progress towards the Green-Griffiths conjecture for higher-dimensional mappings.

Abstract

Motivated by the Green-Griffiths conjecture, we study maximal rank holomorphic maps from $\C^p$ into complex manifolds. When $p>1$ such maps should in principle be more tractable than entire curves. We extend to this setting the jet-bundles techniques introduced by Semple, Green-Griffiths and Demailly. Our main application is the non-existence of maximal rank holomorphic maps from $\C^2$ into the very general degree $d$ hypersurface in $\bP^4$, as soon as $d\geq 93.$