Endomorphisms of hypersurfaces of Fano manifolds of Picard number 1

TL;DR

This paper investigates whether hypersurfaces within Fano manifolds of Picard number 1 admit non-trivial endomorphisms, extending known results from projective spaces to more general Fano manifolds.

Contribution

It generalizes Beauville's result by studying endomorphisms of hypersurfaces in arbitrary Fano manifolds of Picard number 1.

Findings

Supports the conjecture for broader classes of Fano manifolds

Extends the understanding of endomorphisms beyond projective spaces

Provides new insights into the structure of hypersurfaces in Fano manifolds

Abstract

It is conjectured that a Fano manifold of Picard number 1 which is not a projective space admits no endomorphisms of degree bigger than 1. Beauville confirmed this for hypersurfaces of projective space. We study this problem for hypersurfaces of an arbitrary Fano manifold of Picard number 1.