On a Torelli principle for automorphisms of Klein hypersurfaces (Q6571609)

In this paper, for \(d \geq 3\) and \(n \geq 1\) integers, the authors consider smooth hypersurfaces \(X\) of the \((n+1)\)-dimensional complex projective space \({\mathbb P}^{n+1}\). It is a classical result that, for \((n,d) \neq (1,3), (2,4)\), each (regular) automorphism of \(X\) is the restriction of an element of \(\mathrm{PGL}_{n+2}({\mathbb C})\), the group of linear automorphisms of \({\mathbb P}^{n+1}\). The first result of the paper asserts that, for \((n,d) \neq (1,3), (2,4)\) and under certain extra technical conditions, each automorphism of \(X\) is a generalized triangular matrix. The main technique relies on suitable differential operators, previously used by \N\textit{A. Kontogeorgis} [Manuscr. Math. 107, No. 2, 187--205 (2002; Zbl 0998.14019)], \N\textit{B. Poonen} [Finite Fields Appl. 11, No. 2, 230--268 (2005; Zbl 1076.14053)] and by \N\textit{K. Oguiso} and \textit{X. Yu} [Asian J. Math. 23, No. 2, 201--256 (2019; Zbl 1433.14035)].\N\NIn the particular case of the Klein hypersurface of degree \(d\) \N\[\NX_{n,d}=\{x_{0}^{d-1}x_{1}+x_{1}^{d-1}x_{2}+\cdots+x_{n}^{d-1}x_{n+1}+x_{n+1}^{d-1}x_{0}=0\},\N\] \Nthe above result is used to compute its full group of automorphisms, except for the case \((n,d)=(2,4)\). This computation was previously known, due to \textit{T. Harui} [Kodai Math. J. 42, No. 2, 308--331 (2019; Zbl 1433.14024)] and Oguiso-Yu [loc.cit.], for the case \((n,d) \in \{(1,d), (3,5); d \geq 5\}\).\N\NIn the case that \(n,d \geq 3\) are integers such that \(n+2\) and \(p= ((d-1)^{n+2} +1)/d\) are prime integers (\(p\) is called a Wagstaff prime of base \(d-1\)), the authors prove that the middle primitive cohomology group of \(X_{n,d}\) is an extremal polarized Hodge structure. Moreover, they observe that \(\Aut(H^{n}(X, {\mathbb Z})_{prim})/\{\pm1\} \cong \Aut(X_{n,d})\) in any of the following cases: (a) \(d\) divides \(n+3\) or (b) \(d=3\) and \(n \geq 5\). This result is related to the Strong Torelli Principle for hypersurfaces, which is explained in the paper.