Abstract
We give an example of a pseudoconvex domain in a complex manifold whose \(L^2\)-Dolbeault cohomology is non-Hausdorff, yet the domain is Stein. The domain is a smoothly bounded Levi-flat domain in a two complex-dimensional compact complex manifold. The domain is biholomorphic to a product domain in \({\mathbb {C}}^2\), hence Stein. This implies that for \(q>0\), the usual Dolbeault cohomology with respect to smooth forms vanishes in degree \((p,q)\). But the \(L^2\)-Cauchy–Riemann operator on the domain does not have closed range on \((2,1)\)-forms and consequently its \(L^2\)-Dolbeault cohomology is not Hausdorff.
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References
Barrett, D.E.: Biholomorphic domains with inequivalent boundaries. Invent. Math. 85, 373–377 (1986)
Barrett, D.E.: Global convexity properties of some families of three-dimensional compact Levi-flat hypersurfaces. Trans. Am. Math. Soc. 332, 459–474 (1992)
Boas, H.P., Straube, E.J.: Global regularity of the \({\overline{\partial }}\)-Neumann problem: a survey of the \(L^{2}\)-Sobolev theory. In: Proceedings of Several Complex Variables (Berkeley, CA, 1995–1996), Math. Sci. Res. Inst. Publ. vol. 37, pp. 79–111. Cambridge University Press, Cambridge (1999)
Brüning, J., Lesch, M.: Hilbert complexes. J. Funct. Anal. 108, 88–132 (1992)
Cao, J., Shaw, M.-C., Wang, L.: Estimates for the \(\overline{\partial }\)-Neumann problem and nonexistence of \(C^2\)-Levi-flat hypersurfaces in \(\mathbb{CP}^n\). Math. Z. 248, 183–221 (2004)
Chakrabarti, D., Shaw, M.-C.: \(L^2\) Serre duality on domains in complex manifolds and applications. Trans. Am. Math. Soc. 364, 3529–3554 (2012)
Chen, S.-C., Shaw, M.-C.: Partial differential equations in several complex variables. In: Proceedings of AMS/IP Studies in Advanced Mathematics, vol. 19. American Mathematical Society/International Press, Providence/Boston (2001)
Diederich, K., Fornaess, J.E.: Pseudoconvex domains: bounded strictly plurisubharmonic exhaustion functions. Invent. Math. 39, 129–141 (1977)
Diederich, K., Ohsawa, T.: A Levi problem on two-dimensional complex manifolds. Math. Ann. 261, 255–261 (1982)
Diederich, K., Ohsawa, T.: On certain existence questions for pseudoconvex hypersurfaces in complex manifolds. Math. Ann. 323, 397–403 (2002)
Folland, G.B., Kohn, J.J.: The Neumann problem for the Cauchy–Riemann complex. In: Proceedings of Annals of Mathematics Studies, vol. 75. Princeton University Press/University of Tokyo Press, Princeton/Tokyo (1972)
Gunning, R.C., Rossi, H.: Analytic Functions of Several Complex Variables. Prentice-Hall Inc, Englewood Cliffs (1965)
Hörmander, L.: An introduction to complex analysis in several variables, 3rd edn. North-Holland Mathematical Library 7, North-Holland Publishing Co., Amsterdam (1990)
Kazama, H.: \({\overline{\partial }}\)-cohomology of \((H, C)\)-groups. Publ. Res. Inst. Math. Sci. 20, 297–317 (1984)
Kohn, J.J., Nirenberg, L.: An algebra of pseudo-differential operators. Comm. Pure Appl. Math. 18, 269–305 (1965)
Kohn, J.J., Nirenberg, L.: Noncoercive boundary value problems. Comm. Pure Appl. Math. 18, 443–492 (1965)
Kohn, J.J.: Global regularity for \(\overline{\partial }\) on weakly pseudoconvex manifolds. Trans. Am. Math. Soc. 181, 273–292 (1973)
Kohn, J.J., Rossi, H.: On the extension of holomorphic functions from the boundary of a complex manifold. Ann. Math. 81, 451–472 (1965)
Laurent-Thiébaut, C., Shaw, M.-C.: On the Hausdorff property of some Dolbeault cohomology groups. Math. Zeit. 274, 1165–1176 (2013)
Lions, J.-L., Magenes, E.: Non-homogeneous boundary value problems and applications. vol. I. Translated from the French by P. Kenneth. Die Grundlehren der mathematischen Wissenschaften, Band 181. Springer, New York (1972)
Lupacciolu, G.: Holomorphic extension to open hulls. Annali della Scuola Normale Superiore di Pisa 23, 363–382 (1996)
Malgrange, B.: La cohomologie d’une variété analytique complexe à bord pseudo-convexe n’est pas nécessairement séparée. C. R. Acad. Sci. Paris Sér. A-B 280, Aii, A93–A95 (1975)
Ohsawa, T.: A Stein domain with smooth boundary which has a product structure. Publ. Res. Inst. Math. Sci. 18, 1185–1186 (1982)
Ohsawa, T., Sibony, N.: Bounded P.S.H functions and pseudoconvexity in Kähler manifolds. Nagoya Math. J. 149, 1–8 (1998)
Serre, J.-P.: Un théorème de dualité. Comment. Math. Helv. 29, 9–26 (1955)
Shaw, M.-C.: The closed range property for \({\overline{\partial }}\) for domains with pseudoconcave boundary. In: Proceedings of Complex Analysis, Trends in Mathematics. Springer, Basel, pp. 307–320 (2010)
Straube, E.J.: Lectures on the \(L^2\)-Sobolev Theory of the \(\overline{\partial }\)-Neumann Problem. European Mathematical Society, Zürich (2010)
Taylor, M.E.: Partial differential equations. I. Basic theory. Applied Mathematical Sciences, vol. 115. Springer, New York (1996)
Acknowledgments
Debraj Chakrabarti would like to thank K. Sandeep for helpful discussions on fractional Sobolev spaces. Mei-Chi Shaw would like to thank David Barrett for pointing out his paper [1] which inspires the present work. Both authors would like to thank Sophia Vassiliadou for her comments on a previous version of the manuscript, and the referee for numerous helpfull suggestions.
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Debraj Chakrabarti was partially supported by a grant from the Simons Foundation (#316632), the Indo-US Virtual Institute for Mathematical and Statistical Sciences (VI-MSS), and an Early Career internal grant from Central Michigan University. Mei-Chi Shaw was partially supported by National Science Foundation grants DMS-1101415 and DMS-1362175.
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Chakrabarti, D., Shaw, MC. The \(L^2\)-cohomology of a bounded smooth Stein Domain is not necessarily Hausdorff. Math. Ann. 363, 1001–1021 (2015). https://doi.org/10.1007/s00208-015-1193-0
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DOI: https://doi.org/10.1007/s00208-015-1193-0