## Nagoya Mathematical Journal

### On the complement of Levi-flats in K\"{a}hler manifolds of dimension $\ge 3$

Takeo Ohsawa

#### Abstract

Applying the $L^2$ method of solving the $\bar{\partial}$-equation, it is shown that compact K\"{a}hler manifolds of dimension $\geq 3$ admit no Levi flat real analytic hypersurfaces whose complements are Stein.

#### Article information

Source
Nagoya Math. J., Volume 185 (2007), 161-169.

Dates
First available in Project Euclid: 23 March 2007

https://projecteuclid.org/euclid.nmj/1174668920

Mathematical Reviews number (MathSciNet)
MR2301464

Zentralblatt MATH identifier
1141.32015

Subjects
Primary: 32V40: Real submanifolds in complex manifolds

#### Citation

Ohsawa, Takeo. On the complement of Levi-flats in K\"{a}hler manifolds of dimension $\ge 3$. Nagoya Math. J. 185 (2007), 161--169. https://projecteuclid.org/euclid.nmj/1174668920

#### References

• N. Aronszajn, A unique continuation theorem for solutions of elliptic partial differential equations or inequalities of second order, J. Math. Pures Appl., 36 (1957), 235–249.
• J. Cao, M.-C. Shaw, and L. Wang, Estimates for the $\bar{\partial}$-Neumann problem and nonexistence of $C^2$ Levi-flat hypersurfaces in $\mathbb{C}\mathbb{P}^{n}$, Math. Z., 248 (2004), 183–221; Erratum, 223–225.
• J.-P. Demailly, Cohomology of $q$-convex spaces in top degree, Math. Z., 204 (1990), 283–295.
• K. Diederich, and T. Ohsawa, Harmonic mappings and the disc bundles over compact Kähler manifolds, Publ. RIMS, 21 (1985), 819–833.
• H. Grauert, Bemerkenswerte pseudokonvexe Mannigfaltigkeiten, Math. Z., 81 (1963), 377–392
• L. Hörmander, $L^2$ estimates and existence theorems for the $\bar{\partial}$-operator, Acta Math, 113 (1965), 89–152.
• ––––, An Introduction to Complex Analysis in Several Variavbles,2004, North Holland Publishing Company, 1973.
• A. Iordan, On the existence of smooth Levi-flat hypersurfaces in $\mathbb{C}\mathbb{P}^{n}$, Advanced Studies in Pure Mathematics 42, Complex Analysis in Several Variables, pp. 123–126.
• N. Mok, The Serre problem on Riemann surfaces, Math. Ann., 258 (1981), 145–168.
• S. Nemirovski, Stein domains with Levi-flat boundaries on compact complex surfaces, Math. Notes, 66, no.4 (1999), 522–525.
• T. Ohsawa, A reduction theorem for cohomology groups of very strongly $q$ convex Kähler manifolds, Invent. Math. 63 (1981), 335–354. Addendum: Invent. Math., 66 (1982), 391–393.
• ––––, A Stein domain with smooth boundary which has a product structure, Publ. RIMS, 18 (1982), 1185–1186.
• ––––, On the Levi-flats in complex tori of dimension two, Publ. RIMS, 42 (2006), 361–377. Supplement; 379–382, Erratum; preprint.
• ––––, A Levi-flat in a Kummer surface whose complement is strongly pseudoconvex, Osaka J. Math., 43 (2006), 747–750
• T. Ohsawa, and K. Takegoshi, Hodge spectral sequence on pseudoconvex domains, Math. Z., 197 (1988), 1–12.
• Y.T. Siu, Nonexistence of smooth Levi-flat hypersurfaces in complex projective spaces of dimension $\geq 3$, Ann. of Math., 151 (2000), 1217–1243.
• ––––, $\bar{\partial}$-regularity for weakly pseudoconvex domains in compact Hermitian spaces with respect to invariant metrics, Ann. of Math., 156 (2002), 595–621.