On a Torelli Principle for automorphisms of Klein hypersurfaces

TL;DR

This paper refines the differential method to determine automorphism groups of Klein hypersurfaces, introduces extremal polarized Hodge structures, and verifies the Torelli Principle for specific cases, advancing understanding of symmetries in algebraic geometry.

Contribution

It provides effective criteria for automorphisms of hypersurfaces, computes automorphism groups for Klein hypersurfaces, and introduces extremal polarized Hodge structures to analyze automorphisms.

Findings

Automorphisms are given by generalized triangular matrices under certain conditions.

Computed automorphism groups for all Klein hypersurfaces with specified dimensions and degrees.

Confirmed the Torelli Principle for cubic hypersurfaces and some other cases.

Abstract

Using a refinement of the differential method introduced by Oguiso and Yu, we provide effective conditions under which the automorphisms of a smooth degree $d$ hypersurface of $\mathbf{P}^{n+1}$ are given by generalized triangular matrices. Applying this criterion we compute all the remaining automorphism groups of Klein hypersurfaces of dimension $n\geq 1$ and degree $d\geq 3$ with $(n,d)\neq (2,4)$. We introduce the concept of extremal polarized Hodge structures, which are structures that admit an automorphism of large prime order. Using this notion, we compute the automorphism group of the polarized Hodge structure of certain Klein hypersurfaces that we call of Wagstaff type, which are characterized by the existence of an automorphism of large prime order. For cubic hypersurfaces and some other values of $(n,d)$, we show that both groups coincide (up to involution) as predicted by the Torelli Principle.