On a Torelli Principle for automorphisms of Klein hypersurfaces

González-Aguilera, Víctor; Liendo, Alvaro; Montero, Pedro; Villaflor Loyola, Roberto

doi:10.1090/tran/9111

On a Torelli Principle for automorphisms of Klein hypersurfaces
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by Víctor González-Aguilera, Alvaro Liendo, Pedro Montero and Roberto Villaflor Loyola;

Trans. Amer. Math. Soc. 377 (2024), 5483-5511

DOI: https://doi.org/10.1090/tran/9111

Published electronically: June 20, 2024

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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.

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