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].


57R99 Differential topology
53A20 Projective differential geometry
14J70 Hypersurfaces and algebraic geometry
Full Text: DOI Numdam EuDML


[1] E. CARTAN , Sur la déformation projective des surfaces (Ann. Sci. Ecole Norm. Sup., Vol. 37, 1920 , pp. 259-356). Numdam | JFM 47.0656.05 · JFM 47.0656.05
[2] E. CECH and G. FUBINI , Geometria proiettiva differenziale, II (Zanichelli, Bologna, 1927 ). JFM 53.0702.01 · JFM 53.0702.01
[3] E. ČECH and G. FUBINI , Introduction à la géométrie projective différentielle des surfaces (Gauthier-Villars, Paris, 1931 ). Zbl 0002.35101 | JFM 57.0936.01 · Zbl 0002.35101
[4] G. FUBINI , Applicabilitá proiettiva di due superficie (Rend. Circolo matem. di Palermo, XLI, 1916 , pp. 135-162). JFM 46.1098.01 · JFM 46.1098.01
[5] G. FUBINI , Studi relativi all’elemento lineare proiettivo di una ipersuperficie (Rend. Acad. Naz. dei Lincei, 1918 , pp. 99-106). JFM 46.1095.02 · JFM 46.1095.02
[6] G. FUBINI , Il problema della deformazione proiettiva delle ipersuperficie (Rend. Acad. Naz. dei Lincei, Vol. 27, 1918 , pp. 147-155). JFM 46.1095.03 · JFM 46.1095.03
[7] G. FUBINI , Sur les surfaces projectivement applicables (Comptes Rendus Acad. Sci., Vol. 171, 1920 , p. 88). JFM 47.0656.04 · JFM 47.0656.04
[8] P. GRIFFITHS , On Cartan’s method of Lie groups and moving frames as applied to uniqueness and existence questions in differential geometry (Duke J. Math., Vol. 41, 1974 , pp. 775-814). Article | MR 53 #14355 | Zbl 0294.53034 · Zbl 0294.53034 · doi:10.1215/S0012-7094-74-04180-5
[9] P. GRIFFITHS and J. HARRIS , Algebraic geometry and local differential geometry (Ann. Sci. Ecole Norm. Sup., 4e série, t. 12, 1979 , pp. 355-452). Numdam | MR 81k:53004 | Zbl 0426.14019 · Zbl 0426.14019
[10] G. R. JENSEN , Higher order contact of submanifolds of homogeneous spaces (Lecture Notes in Math., Vol. 610, Springer-Verlag, Berlin, 1977 ). MR 58 #18226 | Zbl 0356.53005 · Zbl 0356.53005
[11] G. R. JENSEN , Deformation of submanifolds in homogeneous spaces (J. Diff. Geom., Vol. 16, 1981 , pp. 213-246). MR 83h:53063 | Zbl 0473.53044 · Zbl 0473.53044
[12] C. R. LE BRUN , H-Space with a cosmological constant (Proc. R. Soc. Lond. A., Vol. 380, 1982 , pp. 171-185). MR 83d:83019 | Zbl 0549.53042 · Zbl 0549.53042 · doi:10.1098/rspa.1982.0035
[13] S. KOBAYASHI , Transformation groups in differential geometry (Springer-Verlag, Berlin, 1972 ). MR 50 #8360 | Zbl 0246.53031 · Zbl 0246.53031
[14] P. MALLIAVIN , Géométrie différentielle intrinsèque (Hermann, Paris, 1972 ). MR 57 #13704 | Zbl 0282.53001 · Zbl 0282.53001
[15] E. MUSSO , Sulle condizioni d’integrabilitá delle forme fondamentali di Fubini , (to appear in Rend. Sem. Mat., Univers. Politecn. Torino).
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