Arkiv för Matematik

• Ark. Mat.
• Volume 50, Number 1 (2012), 183-197.

Nonexistence of Levi-degenerate hypersurfaces of constant signature in CPn

Alla Sargsyan

Abstract

Let M be a smooth hypersurface of constant signature in CPn, n≥3. We prove the regularity for on M in bidegree (0,1). As a consequence, we show that there exists no smooth hypersurface in CPn, n≥3, whose Levi form has at least two zero-eigenvalues.

Article information

Source
Ark. Mat., Volume 50, Number 1 (2012), 183-197.

Dates
Revised: 3 April 2011
First available in Project Euclid: 31 January 2017

https://projecteuclid.org/euclid.afm/1485907164

Digital Object Identifier
doi:10.1007/s11512-011-0150-8

Mathematical Reviews number (MathSciNet)
MR2890350

Zentralblatt MATH identifier
1254.32052

Rights

Citation

Sargsyan, Alla. Nonexistence of Levi-degenerate hypersurfaces of constant signature in CP n. Ark. Mat. 50 (2012), no. 1, 183--197. doi:10.1007/s11512-011-0150-8. https://projecteuclid.org/euclid.afm/1485907164

References

• Brinkschulte, J., The Hartogs phenomenon in hypersurfaces with constant signature, Ann. of Math. 183 (2004), 515–535.
• Brinkschulte, J., Nonexistence of higher codimensional Levi-flat CR manifolds in symmetric spaces, J. Reine Angew. Math. 603 (2007), 215–233.
• Cao, J. and Shaw, M.-C., The -Cauchy problem and nonexistence of Lipschitz Levi-flat hypersurfaces in CPn with n≥3, Math. Z. 256 (2007), 175–192.
• Cerveau, D., Minimaux des feuilletages algébriques de CPn, Ann. Inst. Fourier (Grenoble) 43 (1993), 1535–1543.
• Chen, S.-C. and Shaw, M.-C., Partial Differential Equations in Several Complex Variables, Amer. Math. Soc., Providence, RI, 2000.
• Demailly, J.-P., Complex Analytic and Differential Geometry, Notes de cours, Ecole d’été de Mathématiques (Analyse Complexe), Institut Fourier, Grenoble, 1996.
• Freeman, M., Local complex foliation of real submanifolds, Math. Ann. 209 (1974), 1–30.
• Iordan, A., On the nonexistence of smooth Levi-flat hypersurfaces in CPn, in Complex Analysis in Several Variables—Memorial Conference of Kiyoshi Oka’s Centennial Birthday, Adv. Stud. Pure Math. 42, pp. 123–126, Math. Soc. Japan, Tokyo, 2004.
• Lins Neto, A., A note on projective Levi flats and minimal sets of algebraic foliations, Ann. Inst. Fourier (Grenoble) 49 (1999), 1369–1385.
• Matsumoto, K., Pseudoconvex domains of general order and q-convex domains in the complex projective space, J. Math. Kyoto Univ. 33 (1993), 685–695.
• Ohsawa, T., Isomorphism theorems for cohomology groups of weakly 1-complete manifolds, Publ. Res. Inst. Math. Sci. 18 (1982), 191–232.
• Siu, Y.-T., Nonexistence of smooth Levi-flat hypersurfaces in complex projective spaces of dimension ≥3, Ann. of Math. 151 (2000), 1217–1243.
• Siu, Y.-T., -regularity for weakly pseudoconvex domains in compact Hermitian symmetric spaces with respect to invariant metrics, Ann. of Math. 156 (2002), 595–621.