Compactness of the \(\bar{\partial}\)-Neumann operator on singular complex spaces. (English) Zbl 1227.32041
Author’s abstract: Let \(X\) be a Hermitian complex space of pure dimension \(n\). We show that the \(\overline{\partial}\)-Neumann operator on \((p,q)\)-forms is compact at isolated singularities of \(X\) if \(p+q \neq n-1\), \(n\) and \(q \geq 1\). The main step is the construction of compact solution operators for the \(\overline{\partial}\)-equation on such spaces which is based on a general characterization of compactness in function spaces on singular spaces, and that leads also to a criterion for compactness of more general Green operators on singular spaces.
Reviewer: Emil J. Straube (College Station)
References:
| [1] | Alt, H. W., Lineare Funktionalanalysis (1992), Springer-Verlag: Springer-Verlag Berlin · Zbl 0753.46001 |
| [2] | Andreotti, A.; Vesentini, E., Carleman estimates for the Laplace Beltrami equation on complex manifolds, Publ. Math. Inst. Hautes Etudes Sci., 25, 81-130 (1965) |
| [3] | Aroca, J. M.; Hironaka, H.; Vicente, J. L., Desingularization Theorems, Mem. Math. Inst. Jorge Juan, vol. 30 (1977) · Zbl 0366.32009 |
| [4] | 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) · Zbl 0896.14006 |
| [5] | Coltoiu, M.; Mihalache, N., Strongly plurisubharmonic exhaustion functions on 1-convex spaces, Math. Ann., 270, 63-68 (1985) · Zbl 0533.32009 |
| [6] | Diederich, K.; Fornæss, J. E.; Vassiliadou, S., Local \(L^2\) results for \(\overline{\partial}\) on a singular surface, Math. Scand., 92, 269-294 (2003) · Zbl 1033.32005 |
| [7] | Folland, G. B.; Kohn, J. J., The Neumann Problem for the Cauchy-Riemann Complex, Ann. of Math. Stud., vol. 75 (1972), Princeton University Press: Princeton University Press Princeton · Zbl 0247.35093 |
| [8] | Fornæss, J. E., \(L^2\) results for \(\overline{\partial}\) in a conic, (International Symposium, Complex Analysis and Related Topics. International Symposium, Complex Analysis and Related Topics, Cuernavaca. International Symposium, Complex Analysis and Related Topics. International Symposium, Complex Analysis and Related Topics, Cuernavaca, Oper. Theory Adv. Appl. (1999), Birkhäuser) |
| [9] | Fornæss, J. E.; Øvrelid, N.; Vassiliadou, S., Semiglobal results for \(\overline{\partial}\) on a complex space with arbitrary singularities, Proc. Amer. Math. Soc., 133, 8, 2377-2386 (2005) · Zbl 1073.32022 |
| [10] | Fornæss, J. E.; Øvrelid, N.; Vassiliadou, S., Local \(L^2\) results for \(\overline{\partial} \): the isolated singularities case, Internat. J. Math., 16, 4, 387-418 (2005) · Zbl 1081.32024 |
| [11] | Fox, J.; Haskell, P., Hodge decompositions and Dolbeault complexes on complex surfaces, Trans. Amer. Math. Soc., 343, 765-778 (1994) · Zbl 0806.58001 |
| [12] | Gaffney, M. P., Hilbert space methods in the theory of harmonic integrals, Trans. Amer. Math. Soc., 78, 426-444 (1955) · Zbl 0064.34303 |
| [13] | Gansberger, K., On the resolvent of the Dirac operator in \(R^2\) |
| [14] | Haskell, P., \(L^2\)-Dolbeault complexes on singular curves and surfaces, Proc. Amer. Math. Soc., 107, 2, 517-526 (1989) · Zbl 0684.58040 |
| [15] | Haslinger, F., Compactness for the \(\overline{\partial} \)-Neumann problem - a functional analysis approach, ESI-Preprint 2208 · Zbl 1217.32014 |
| [16] | Hauser, H., The Hironaka theorem on resolution of singularities, Bull. Amer. Math. Soc. (N.S.), 40, 3, 323-403 (2003) · Zbl 1030.14007 |
| [17] | Hefer, T.; Lieb, I., On the compactness of the \(\overline{\partial} \)-Neumann operator, Ann. Fac. Sci. Toulouse Math. (6), 9, 3, 415-432 (2000) · Zbl 1017.32025 |
| [18] | Hironaka, H., Resolution of singularities of an algebraic variety over a field of characteristic zero: I, II, Ann. of Math. (2), 79, 109-326 (1964) · Zbl 0122.38603 |
| [19] | Hörmander, L., \(L^2\)-estimates and existence theorems for the \(\overline{\partial} \)-operator, Acta Math., 113, 89-152 (1965) · Zbl 0158.11002 |
| [20] | Hörmander, L., An Introduction to Complex Analysis in Several Variables (1966), Van Nostrand: Van Nostrand Princeton · Zbl 0138.06203 |
| [21] | Kashiwara, M.; Kawai, T., The Poincaré lemma for varieties of polarized Hodge structure, Publ. Res. Inst. Math. Sci., 23, 345-407 (1987) · Zbl 0629.14005 |
| [22] | Kohn, J. J., Harmonic integrals on strongly pseudoconvex manifolds I, Ann. of Math., 78, 112-148 (1963) · Zbl 0161.09302 |
| [23] | Kohn, J. J., Harmonic integrals on strongly pseudoconvex manifolds II, Ann. of Math., 79, 450-472 (1964) · Zbl 0178.11305 |
| [24] | Kohn, J. J.; Nirenberg, L., Non-coercive boundary value problems, Comm. Pure Appl. Math., 18, 443-492 (1965) · Zbl 0125.33302 |
| [25] | Lieb, I.; Michel, J., The Cauchy-Riemann Complex, Integral Formulae and Neumann Problem (2002), Vieweg: Vieweg Braunschweig/Wiesbaden · Zbl 0994.32002 |
| [26] | Nagase, M., Remarks on the \(L^2\)-Dolbeault cohomology groups of singular algebraic surfaces and curves, Publ. Res. Inst. Math. Sci., 26, 5, 867-883 (1990) · Zbl 0726.14014 |
| [27] | Ohsawa, T., Hodge spectral sequence on compact Kähler spaces, Publ. Res. Inst. Math. Sci., 23, 265-274 (1987) · Zbl 0626.32029 |
| [28] | Ohsawa, T.; Takegoshi, K., On the extension of \(L^2\)-holomorphic functions, Math. Z., 195, 2, 197-204 (1987) · Zbl 0625.32011 |
| [29] | Øvrelid, N.; Vassiliadou, S., Solving \(\overline{\partial}\) on product singularities, Complex Var. Elliptic Equ., 51, 3, 225-237 (2006) · Zbl 1127.32007 |
| [30] | Øvrelid, N.; Vassiliadou, S., Some \(L^2\) results for \(\overline{\partial}\) on projective varieties with general singularities, Amer. J. Math., 131, 129-151 (2009) · Zbl 1172.32013 |
| [31] | Pardon, W., The \(L^2-\overline{\partial} \)-cohomology of an algebraic surface, Topology, 28, 2, 171-195 (1989) · Zbl 0682.32024 |
| [32] | Pardon, W.; Stern, M., \(L^2-\overline{\partial} \)-cohomology of complex projective varieties, J. Amer. Math. Soc., 4, 3, 603-621 (1991) · Zbl 0751.14011 |
| [33] | 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) · Zbl 0960.14009 |
| [34] | Rudin, W., Functional Analysis, International Series, Pure Appl. Math. (1991), McGraw-Hill: McGraw-Hill New York · Zbl 0867.46001 |
| [35] | Ruppenthal, J., About the \(\overline{\partial} \)-equation at isolated singularities with regular exceptional set, Internat. J. Math., 20, 4, 459-489 (2009) · Zbl 1180.32015 |
| [36] | Ruppenthal, J., The \(\overline{\partial} \)-equation on homogeneous varieties with an isolated singularity, Math. Z., 263, 447-472 (2009) · Zbl 1187.32007 |
| [37] | J. Ruppenthal, \(L^2\overline{\partial}\) arXiv:1004.0396; J. Ruppenthal, \(L^2\overline{\partial}\) arXiv:1004.0396 |
| [38] | Ruppenthal, J.; Zeron, E. S., An explicit \(\overline{\partial} \)-integration formula for weighted homogeneous varieties, Michigan Math. J., 58, 321-337 (2009) · Zbl 1214.32017 |
| [39] | Ruppenthal, J.; Zeron, E. S., An explicit \(\overline{\partial} \)-integration formula for weighted homogeneous varieties II, forms of higher degree, Michigan Math. J., 59, 4, 283-295 (2010) · Zbl 1513.32048 |
| [40] | Siu, Y.-T., Analyticity of sets associated to Lelong numbers and the extension of closed positive currents, Invent. Math., 27, 53-156 (1974) · Zbl 0289.32003 |
| [41] | Siu, Y.-T., Invariance of plurigenera, Invent. Math., 134, 3, 661-673 (1998) · Zbl 0955.32017 |
| [42] | Straube, E., Lectures on the \(L^2\)-Sobolev Theory of the \(\overline{\partial} \)-Neumann Problem, ESI Lect. Math. Phys. (2010), European Mathematical Society (EMS): European Mathematical Society (EMS) Zürich · Zbl 1247.32003 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
