×

Nonexistence of Levi-degenerate hypersurfaces of constant signature in \(\mathrm {CP}^n\). (English) Zbl 1254.32052

Summary: Let \(M\) be a smooth hypersurface of constant signature in \(\mathbb {CP}^n\), \(n\geq 3\). We prove the regularity for \(\overline \partial_b\) on \(M\) in bidegree \((0,1)\). As a consequence, we show that there exists no smooth hypersurface in \(\mathbb {CP}^n\), \(n\geq 3\), whose Levi form has at least two zero-eigenvalues.

MSC:

32V40 Real submanifolds in complex manifolds
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Brinkschulte, J., The Hartogs phenomenon in hypersurfaces with constant signature, Ann. of Math. 183 (2004), 515–535. · Zbl 1121.32017 · doi:10.1007/s10231-004-0103-y
[2] Brinkschulte, J., Nonexistence of higher codimensional Levi-flat CR manifolds in symmetric spaces, J. Reine Angew. Math. 603 (2007), 215–233. · Zbl 1125.32011
[3] Cao, J. and Shaw, M.-C., The -Cauchy problem and nonexistence of Lipschitz Levi-flat hypersurfaces in CP n with n, Math. Z. 256 (2007), 175–192. · Zbl 1124.32017 · doi:10.1007/s00209-006-0064-5
[4] Cerveau, D., Minimaux des feuilletages algébriques de CP n , Ann. Inst. Fourier (Grenoble) 43 (1993), 1535–1543. · Zbl 0803.32018 · doi:10.5802/aif.1382
[5] Chen, S.-C. and Shaw, M.-C., Partial Differential Equations in Several Complex Variables, Amer. Math. Soc., Providence, RI, 2000.
[6] Demailly, J.-P., Complex Analytic and Differential Geometry, Notes de cours, Ecole d’été de Mathématiques (Analyse Complexe), Institut Fourier, Grenoble, 1996.
[7] Freeman, M., Local complex foliation of real submanifolds, Math. Ann. 209 (1974), 1–30. · doi:10.1007/BF01432883
[8] Iordan, A., On the nonexistence of smooth Levi-flat hypersurfaces in CP n , 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. · Zbl 1073.32019
[9] Lins Neto, A., A note on projective Levi flats and minimal sets of algebraic foliations, Ann. Inst. Fourier (Grenoble) 49 (1999), 1369–1385. · Zbl 0963.32022 · doi:10.5802/aif.1721
[10] Matsumoto, K., Pseudoconvex domains of general order and q-convex domains in the complex projective space, J. Math. Kyoto Univ. 33 (1993), 685–695. · Zbl 0797.32013
[11] Ohsawa, T., Isomorphism theorems for cohomology groups of weakly 1-complete manifolds, Publ. Res. Inst. Math. Sci. 18 (1982), 191–232. · Zbl 0526.32016 · doi:10.2977/prims/1195184021
[12] Siu, Y.-T., Nonexistence of smooth Levi-flat hypersurfaces in complex projective spaces of dimension , Ann. of Math. 151 (2000), 1217–1243. · Zbl 0980.53065 · doi:10.2307/121133
[13] 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. · Zbl 1030.53071 · doi:10.2307/3597199
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.