Abstract
Let X be a regular irreducible variety in \({\mathbb{CP}^{n-1}}\), Y the associated homogeneous variety in \({\mathbb{C}^n}\), and N the restriction of the universal bundle of \({\mathbb{CP}^{n-1}}\) to X. In the present paper, we compute the obstructions to solving the \({\overline{\partial}}\)-equation in the L p-sense on Y for 1 ≤ p ≤ ∞ in terms of cohomology groups \({H^q(X,\mathcal {O}(N^\mu))}\). That allows to identify obstructions explicitly if X is specified more precisely, for example if it is equivalent to \({\mathbb{CP}^1}\) or an elliptic curve.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Abhyankar S.: Concepts of order and rank on a complex space, and a condition for normality. Math. Ann. 141, 171–192 (1960)
Acosta F., Zeron E.S.: Hölder estimates for the \({\overline{\partial}}\)-equation on surfaces with simple singularities. Bol. Soc. Mat. Mexicana (3) 12(2), 193–204 (2006)
Acosta F., Zeron E.S.: Hölder estimates for the \({\overline{\partial}}\)-equation on surfaces with singularities of the type E 6 and E 7. Bol. Soc. Mat. Mexicana (3) 13(1), (2007)
Alt H.W.: Lineare Funktional analysis. Springer, Berlin (1992)
Aroca, J.M., Hironaka, H., Vicente, J.L.: Desingularization theorems. Mem. Math. Inst. Jorge Juan 30, Madrid, 1977
Bierstone E., Milman P.: Canonical desingularization in characteristic zero by blowing-up the maximum strata of a local invariant. Invent. Math. 128(2), 207–302 (1997)
Cheeger J., Goresky M., MacPherson R.: L 2-cohomology and intersection homology of singular algebraic varieties. Ann. Math. Stud. 102, 303–340 (1982)
Diederich K., Fornæss J.E., Vassiliadou S.: Local L 2 results for \({\overline{\partial}}\) on a singular surface. Math. Scand. 92, 269–294 (2003)
Fornæss, J.E.: L 2 results for \({\overline{\partial}}\) in a conic. In: International symposium, complex analysis and related topics, Cuernavaca. Operator theory: advances and applications (Birkhauser, 1999)
Fornæss J.E., Gavosto E.A.: The Cauchy Riemann equation on singular spaces. Duke Math. J. 93, 453–477 (1998)
Fornæss J.E., Øvrelid N., Vassiliadou S.: Semiglobal results for \({\overline{\partial}}\) on a complex space with arbitrary singularities, Proc. Am. Math. Soc. 133(8), 2377–2386 (2005)
Fornæss J.E., Øvrelid N., Vassiliadou S.: Local L 2 results for \({\overline{\partial}}\): the isolated singularities case. Int. J. Math. 16(4), 387–418 (2005)
Grauert H.: Über Modifikationen und exzeptionelle analytische Mengen. Math. Ann. 146, 331–368 (1962)
Grauert H., Remmert R.: Theory of Stein spaces. Springer, Berlin (1979)
Hauser H.: The Hironaka theorem on resolution of singularities. Bull. (New Ser.) Am. Math. Soc. 40(3), 323–403 (2003)
Hefer, T.: Optimale Regularitätssätze für die \({\overline{\partial}}\)-Gleichung auf gewissen konkaven und konvexen Modellgebieten. Bonner Math. Schr. 301 (1997)
Henkin G.M, Leiterer J.: Theory of functions on complex mannifolds. Monogr. Math. 79, Birkhäuser Verlag, Basel (1984)
Krantz S.G.: Optimal Lipschitz and L p regularity for the equation \({\overline{\partial} u=f}\) on strongly pseudoconvex domains. Math. Ann. 219, 233–260 (1976)
Lieb I., Michel J.: The Cauchy–Riemann complex, integral formulae and Neumann problem. Vieweg, Braunschweig/Wiesbaden (2002)
Narasimhan R.: The Levi problem for complex spaces (II). Math. Ann. 146, 195–216 (1962)
Merker, J., Porten, E.: The Hartogs’ extension theorem on (n − 1)-complete complex spaces, preprint, arXiv:0704.3216
Ohsawa T.: Cheeger–Goresky–MacPherson’s conjecture for the varieties with isolated singularities. Math. Z. 206, 219–224 (1991)
Øvrelid N., Vassiliadou S.: Solving \({\overline{\partial}}\) on product singularities, complex. Var Ellipitic Equ. 51(3), 225–237 (2006)
Pardon W.L., Stern M.A.: L 2-\({\overline{\partial}}\)-cohomology of complex projective varieties. J. Am. Math. Soc. 4, 603–621 (1991)
Pardon W., Stern M.: Pure Hodge structure on the L 2-cohomology of varieties with isolated singularities. J. Reine Angew. Math. 533, 55–80 (2001)
Range R.M.: Holomorphic functions and integral representations in several complex variables. (Graduate Texts in Mathematics, Bd. 108), Springer, New York (1986)
Ruppenthal, J.: Zur Regularität der Cauchy-Riemannschen Differentialgleichungen auf komplexen Räumen, Bonner Math. Schr. 380 (2006)
Ruppenthal, J.: About the \({\overline{\partial}}\)-equation at isolated singularities with regular exceptional set, preprint 2007, arXiv:0803.0152. Int. J. Math. (to appear)
Ruppenthal, J.: A \({\overline{\partial}}\)-theoretical proof of Hartogs’ extension theorem on Stein spaces with isolated singularities, preprint 2007, arXiv:0803.0137. J. Geom. Anal. (to appear)
Ruppenthal, J.: The Dolbeault complex with weights according to normal crossings. Math. Z. (2008). doi:10.1007/s00209-008-0424-4
Ruppenthal, J., Zeron, E.S.: An explicit \({\overline{\partial}}\)-integration formula for weighted homogeneous varieties, preprint 2008, arXiv:0803:0136, Mich. Math. J. (to appear)
Scheja G.: Riemannsche Hebbarkeitssätze für Cohomologieklassen. Math. Ann. 144, 345–360 (1961)
Scheja G.: Eine Anwendung Riemannscher Hebbarkeitssätze für analytische Cohomologieklassen. Arch. Math. 12, 341–348 (1961)
Solis, M., Zeron, E.S.: Hölder estimates for the \({\overline{\partial}}\)-equation on singular quotient varieties. Bol. Soc. Math. Mexicana (3) (to appear)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Ruppenthal, J. The \({\overline{\partial}}\)-equation on homogeneous varieties with an isolated singularity. Math. Z. 263, 447–472 (2009). https://doi.org/10.1007/s00209-008-0425-3
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00209-008-0425-3