Abstract
The purpose of this paper is to study holomorphic approximation and approximation of \(\overline{\partial }\)-closed forms in complex manifolds of complex dimension \(n\ge 1\). We consider extensions of the classical Runge theorem and the Mergelyan property to domains in complex manifolds for the \({{\mathcal {C}}}^\infty \)-smooth and the \(L^2\) topology. We characterize the Runge or Mergelyan property in terms of certain Dolbeault cohomology groups and some geometric sufficient conditions are given.
Similar content being viewed by others
References
Chakrabarti, D., Shaw, M.-C.: \({L}^2\) Serre duality on domains in complex manifolds and applications. Trans. A.M.S 364, 3529–3554 (2012)
Chen, S.-C., Shaw, M.-C.: Partial differential equations in several complex variables. In: Studies in Advanced Math, vol. 19. American Mathematical Society, International Press, Providence (2001)
Chirka, E.M., Stout, E.L.: Removable singularities in the boundary. Contrib Complex Anal Anal Geom Asp Math E26, 43–104 (1994)
Fornaess, J.E., Forstneric, F., Wold, E.F.: Holomorphic approximation: the legacy of Weierstrass, Runge, Oka–Weil, and Mergelyan. arXiv:1802.03924v2 [math.CV]
Fu, S., Laurent-Thiébaut, C., Shaw, M.C.: Hearing pseudoconvexity in lipschitz domains with holes via \({\overline{\partial }}\). Math. Zeit. 287, 1157–1181 (2017)
Henkin, G.M., Leiterer, J.: Andreotti–Grauert theory by integral formulas, Progress in Math, vol. 74. Birkhaüser, Basel, Boston, Berlin (1988)
Hörmander, L.: \(L^2\) Estimates and Existence Theorems for the \({\overline{\partial }}\) Operator. Acta Math. 113, 89–152 (1965)
Hörmander, L.: An introduction to complex analysis in several complex variables. Van Nostrand, Princeton (1990)
Laurent-Thiébaut, C.: Théorie des fonctions holomorphes de plusieurs variables, Savoirs actuels. InterEditions/CNRS Editions, Paris (1997)
Laurent-Thiébaut, C.: Sur l’équation de Cauchy–Riemann tangentielle dans une calotte strictement pseudoconvexe. Int. J Math. 16, 1063–1079 (2005)
Laurent-Thiébaut, C., Leiterer, J.: On the Hartogs–Bochner extension phenomenon for differential forms. Math. Ann. 284, 103–119 (1989)
Laurent-Thiébaut, C., Shaw, M.C.: On the Hausdorff property of some Dolbeault cohomology groups. Math. Zeitschrift 274, 1165–1176 (2013)
Laurent-Thiébaut, C., Shaw, M.C.: Solving \({\overline{\partial }}\) with prescribed support on hartogs triangles in \({\mathbb{C}}^2\) and \({\mathbb{CP}}^{2}\). Trans. A.M.S. 371, 6531–6546 (2019)
Lupacciolu, G.: Characterization of removable sets in strongly pseudoconvex boundaries. Ark. Mat. 32, 455–473 (1994)
Malgrange, B.: Existence et approximation des solutions des équations aux dérivées partielles et des équations de convolution. Ann. Inst. Fourier 6, 271–355 (1956)
Sambou, S.: Résolution du \({\overline{\partial }}_{b}\) pour les courants prolongeables définis dans un anneau. Ann. Fac. Sci. Toulouse 11, 105–129 (2002)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
The first author would like to thank the university of Notre Dame for its support during her stay in April 2019. The second author was partially supported by National Science Foundation Grant DMS-1700003.
Rights and permissions
About this article
Cite this article
Laurent-Thiébaut, C., Shaw, MC. Holomorphic approximation via Dolbeault cohomology. Math. Z. 296, 1027–1047 (2020). https://doi.org/10.1007/s00209-020-02470-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00209-020-02470-3