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Rigidity of hypersurfaces in complex projective space. (English) Zbl 0829.57021

Let \(X\) be a connected complex manifold of dimension \(n\) and \(f, \widehat f : X \to \mathbb{C} P^{n + 1}\) be holomorphic immersions such that none of \(\text{det} \varphi\), \(\text{det} \widehat \varphi\), \(\psi\) and \(\widehat \psi\) is identically zero on \(X\), where \(\varphi\) (resp. \(\widehat \varphi)\) is the Fubini holomorphic symmetric quadratic form and \(\psi\) (resp. \(\widehat \psi)\) is the Fubini holomorphic symmetric cubic form associated to a first order frame field along \(f\) (resp. \(\widehat f)\) [see G. Fubini, Studi relativi all’elemento lineare proiettivo di una ipersuperficie, Rend. Acad. Naz. dei Lincei, V. Ser. 27, 99-106 (1918; JFM 46.1095.02)]. Fubini’s theorem states that if, calculated with respect to third order frame fields, \(\psi/ \varphi = \widehat \psi/ \widehat \varphi\) at every point of \(X\), then \(f(X)\) is projectively congruent to \(\widehat f(X)\), that is there is an element \(a \in\text{SL}(n + 2, \mathbb{C})\) such that \(af(X) = \widehat f(X)\). For \(n = 2\), there exist satisfactory proofs of this theorem given by Fubini itself [Applicabilita proiettiva di due superficie, Rend. Circ. Mat. Palermo 41, 135-162 (1916; JFM 46.1098.01)] and by E. Cartan [Sur la déformation projective des surfaces, Ann. Sci. Éc. Norm. Supér., III. Sér. 37, 259-356 (1920; JFM 47.0656.05)]. In the case when \(n > 2\), Fubini’s original proofs [Rend. Acad. Naz. dei Lincei, V. Ser. 27 (loc. cit.) and Il problema della deformazione proiettiva delle ipersuperficie, ibid. 147-155 (1918; JFM 46.1095.03)] are unsatisfactory. Using Cartan’s method of moving frames the authors give a direct elementary proof of Fubini’s theorem when \(N > 2\). Their proof is also constructive in the sense that it gives an algebraic procedure, involving only the diagonalization of a symmetric bilinear form and the solution of linear equations, by which one can find the projective group element which brings the one hypersurface into congruence with the other. The authors’ proof works also in the real case without change, except that one must assume that certain zero divisors are sufficiently thin that the complement of their union is a connected, dense, open subset of \(X\). A readable proof that uses a normalization of the forms which is valid only in the real case appears also in the book of G. Fubini and E. Čech [Geometria proiettiva differenziale, Vol. II (1927; JFM 53.0702.01) pp. 605-629].

MSC:

57R99 Differential topology
53A20 Projective differential geometry
14J70 Hypersurfaces and algebraic geometry
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References:

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