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.


32V40 Real submanifolds in complex manifolds
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[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
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