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Levi problem in complex manifolds

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Abstract

Let U be a pseudoconvex open set in a complex manifold M. When is U a Stein manifold? There are classical counter examples due to Grauert, even when U has real-analytic boundary or has strictly pseudoconvex points. We give new criteria for the Steinness of U and we analyze the obstructions. The main tool is the notion of Levi-currents. They are positive \({\partial \overline{\partial }}\)-closed currents T of bidimension (1, 1) and of mass 1 directed by the directions where all continuous psh functions in U have vanishing Levi-form. The extremal ones, are supported on the sets where all continuous psh functions are constant. We also construct under geometric conditions, bounded strictly psh exhaustion functions, and hence we obtain Donnelly–Fefferman weights. To any infinitesimally homogeneous manifold, we associate a foliation. The dynamics of the foliation determines the solution of the Levi-problem. Some of the results can be extended to the context of pseudoconvexity with respect to a Pfaff-system.

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References

  1. Berndtsson, B: \(L^2\)-methods for the \(\bar{\partial }\)-equation. Preprint http://www.math.chalmers.se/~bob/

  2. Berndtsson, B., Sibony, N.: The \({\overline{\partial }}\)-equation on a positive current. Invent. Math. 147(2), 371–428 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  3. Diederich, K., Fornaess, J.E.: Pseudoconvex domains: bounded strictly plurisubharmonic exhaustion functions. Invent. Math. 39(2), 129–141 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  4. Diederich, K., Fornaess, J.E.: Pseudoconvex domains with real-analytic boundary. Ann. Math. (2) 107(2), 371–384 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  5. Diederich, K., Fornaess, J.E.: A smooth pseudoconvex domain without pseudoconvex exhaustion. Manuscri. Math. 39(1), 119–123 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  6. Diederich, K., Ohsawa, T.: A Levi problem on two-dimensional complex manifolds. Math. Ann. 261(2), 255–261 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  7. Donnelly, H., Fefferman, C.: \(L^2\)-cohomology and index theorem for the Bergman metric. Ann. Math. (2) 118(3), 593–618 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  8. Dinh, T.-C., Sibony, N.: Pull-back currents by holomorphic maps. Manuscri. Math. 123(3), 357–371 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  9. Dinh, T.-C., Sibony, N.: Dynamics in several complex variables: endomorphisms of projective spaces and polynomial-like mappings. In: Lecture Notes in Math., 1998, pp. 165–294. Springer, Berlin (2010)

  10. Elencwajg, G.: Pseudo-convexité locale dans les variété kählériennes. Ann. Inst. Fourier (Grenoble) 25(2), 295–314 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  11. Fornaess, J.E., Sibony, N.: Oka’s inequality for currents and applications. Math. Ann. 301(3), 399–419 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  12. Fornaess, J.E., Sibony, N.: Riemann surface laminations with singularities. J. Geom. Anal. 18(2), 400–442 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  13. Grauert, H.: On Levi’s problem and the imbedding of real-analytic manifolds. Ann. Math. (2) 68, 460–472 (1958)

    Article  MathSciNet  MATH  Google Scholar 

  14. Grauert, H.: Bemerkenswerte pseudokonvexe Mannigfaltigkeiten. Math. Z. 81, 377–391 (1963)

    Article  MathSciNet  MATH  Google Scholar 

  15. Grauert, H.: Kantenkohomologie. Compos. Math. 44(1–3), 79–101 (1981)

    MathSciNet  MATH  Google Scholar 

  16. Hirschowitz, A.: Pseudoconvexité au-dessus d’espaces plus ou moins homogènes. Invent. Math. 26, 303–322 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  17. Hirschowitz, A.: Le problème de Lévi pour les espaces homogènes. Bull. Soc. Math. France 103(2), 191–201 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  18. Hörmander, L.: An Introduction to Complex Analysis in Several Variables, 3rd edn. North Holland, Amsterdam (1988)

    MATH  Google Scholar 

  19. Kaufmann, L.: Self-intersection of foliated cycles on complex manifolds. arXiv:1602.07238

  20. Michel, D.: Sur les ouverts pseudo-convexes des espaces homogènes. C. R. Acad. Sci. Paris Sér. A B 283(10), 779–782 (1976)

    MathSciNet  MATH  Google Scholar 

  21. Narasimhan, R.: The Levi problem for complex spaces. Math. Ann. II 146, 195–216 (1962)

    Article  MathSciNet  MATH  Google Scholar 

  22. Narasimhan, R.: The Levi problem and pseudo-convex domains: a survey. Enseign. Math. (2) 24(3–4), 161–172 (1978)

    MathSciNet  MATH  Google Scholar 

  23. Ohsawa, T., Sibony, N.: Bounded p.s.h. functions and pseudoconvexity in Kähler manifold. Nagoya Math. J. 149, 1–8 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  24. Ohsawa, T.: A survey of Levi flat hypersurfaces. In: Proceedings of the Abel Symposium 2013, NTNU, Trondheim

  25. Peternell, T.: Pseudoconvexity, the Levi problem and vanishing theorems. Complex analysis. Several variables, 7 Encyclopedia of Mathematical Sciences, vol. 74, pp 285–329. Springer (1994)

  26. Richberg, R.: Stetige streng pseudokonvexe Funktionen. Math. Ann. 175, 257–286 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  27. Sibony, N.: Pfaff systems, currents and hulls. Math. Z. 285, 1107–1123 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  28. Sibony, N.: Quelques problèmes de prolongement de courants en analyse complexe. Duke Math. J. 52(1), 157–197 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  29. Siu, Y.T.: Pseudoconvexity and the problem of Levi. Bull. Am. Math. Soc. 84(4), 481–512 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  30. Skoda, H.: Prolongement des courants positifs, fermés de masse finie. Invent. Math. 66, 361–376 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  31. Sullivan, D.: Cycles for the dynamical study of foliated manifolds in complex manifolds. Invent. Math. 36, 225–255 (1975)

    Article  MATH  Google Scholar 

  32. Takeuchi, A.: Domaines pseudoconvexes infinis et la métrique riemannienne dans un espace projectif. J. Math. Soc. Japan 16, 159–181 (1964)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Université Paris-Sud, Orsay, France

    Nessim Sibony

  2. Korea Institute For Advanced Studies, Seoul, Korea

    Nessim Sibony

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  1. Nessim Sibony
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Correspondence to Nessim Sibony.

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In memory of Raghavan Narasimhan.

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Sibony, N. Levi problem in complex manifolds. Math. Ann. 371, 1047–1067 (2018). https://doi.org/10.1007/s00208-017-1539-x

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