Duke Mathematical Journal

L2-theory for the ¯-operator on compact complex spaces

J. Ruppenthal

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


Let X be a singular Hermitian complex space of pure dimension n. We use a resolution of singularities to give a smooth representation of the L2-¯-cohomology of (n,q)-forms on X. The central tool is an L2-resolution for the Grauert–Riemenschneider canonical sheaf KX. As an application, we obtain a Grauert–Riemenschneider-type vanishing theorem for forms with values in almost positive line bundles. If X is a Gorenstein space with canonical singularities, then we also get an L2-representation of the flabby cohomology of the structure sheaf OX. To understand also the L2-¯-cohomology of (0,q)-forms on X, we introduce a new kind of canonical sheaf, namely, the canonical sheaf of square-integrable holomorphic n-forms with some (Dirichlet) boundary condition at the singular set of X. If X has only isolated singularities, then we use an L2-resolution for that sheaf and a resolution of singularities to give a smooth representation of the L2-¯-cohomology of (0,q)-forms.

Article information

Duke Math. J., Volume 163, Number 15 (2014), 2887-2934.

First available in Project Euclid: 1 December 2014

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 32J25: Transcendental methods of algebraic geometry [See also 14C30] 32C35: Analytic sheaves and cohomology groups [See also 14Fxx, 18F20, 55N30] 32W05: $\overline\partial$ and $\overline\partial$-Neumann operators

Cauchy–Riemann equations $L2$-theory singular complex spaces resolution of singularities canonical sheaves Gorenstein singularities


Ruppenthal, J. $L^{2}$ -theory for the $\overline {\partial }$ -operator on compact complex spaces. Duke Math. J. 163 (2014), no. 15, 2887--2934. doi:10.1215/0012794-2838545. https://projecteuclid.org/euclid.dmj/1417442575

Export citation


  • [AV] A. Andreotti and E. Vesentini, Carleman estimates for the Laplace-Beltrami equation on complex manifolds, Publ. Math. Inst. Hautes Études Sci. 25 (1965), 81–130.
  • [AHL] J. M. Aroca, H. Hironaka, and J. L. Vicente, Desingularization theorems, Mem. Mat. Inst. Jorge Juan 30, Consejo Superior de Investigaciones Científicas, Madrid, 1977.
  • [BS] B. Berndtsson and N. Sibony, The $\overline{\partial } $-equation on a positive current, Invent. Math. 147 (2002), 371–428.
  • [BM] E. Bierstone and P. D. Milman, Canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant, Invent. Math. 128 (1997), 207–302.
  • [CGM] J. Cheeger, M. Goresky, and R. MacPherson, “$L^{2}$-cohomology and intersection homology of singular algebraic varieties” in Seminar on Differential Geometry, Ann. of Math. Stud. 102, Princeton Univ. Press, Princeton, 1982, 303–340.
  • [D1] J.-P. Demailly, Estimations $L^{2}$ pour l’opérateur $\overline{\partial } $ d’un fibré vectoriel holomorphe semi-positif au-dessus d’une variété kählérienne complète, Ann. Sci. Éc. Norm. Supér. (4) 15 (1982), 457–511.
  • [D2] J.-P. Demailly, “$L^{2}$ Hodge theory and vanishing theorems” in Introduction to Hodge Theory, SMF/AMS Texts Monogr. 8, Amer. Math. Soc., Providence, 2002, 1–95.
  • [D3] J.-P. Demailly, Complex Analytic and Differential Geometry, Institut Fourier, Université de Grenoble I, Saint-Martin d’Hères, France, 2012, http://www-fourier.ujf-grenoble.fr/~demailly/manuscripts/agbook.pdf.
  • [DF] H. Donelly and C. Fefferman, $L^{2}$-cohomology and index theorem for the Bergman metric, Ann. of Math. (2) 118 (1983), 593–618.
  • [FK] G. B. Folland and J. J. Kohn, The Neumann problem for the Cauchy-Riemann complex, Ann. of Math. Stud. 75, Princeton Univ. Press, Princeton, 1972.
  • [FOV] J. E. Fornæss, N. Øvrelid, and S. Vassiliadou, Local $L^{2}$ results for $\overline{\partial } $: The isolated singularities case, Internat. J. Math. 16 (2005), 387–418.
  • [GM1] M. Goresky and R. MacPherson, Intersection homology theory, Topology 19 (1980), 135–162.
  • [GM2] M. Goresky and R. MacPherson, Intersection homology, II, Invent. Math. 72 (1983), 77–129.
  • [GR] H. Grauert and O. Riemenschneider, Verschwindungssätze für analytische Kohomologiegruppen auf komplexen Räumen, Invent. Math. 11 (1970), 263–292.
  • [H1] H. Hauser, The Hironaka theorem on resolution of singularities (or: A proof we always wanted to understand), Bull. Amer. Math. Soc. (N.S.) 40, (2003), 323–403.
  • [HL] G. M. Henkin and J. Leiterer, Theory of Functions on Complex Manifolds, Monogr. Math. 79, Birkhäuser, Basel, 1984.
  • [H2] H. Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero, I, Ann. of Math. (2) 79 (1964), 109–203; II, 205–326.
  • [H3] L. Hörmander, $L^{2}$ estimates and existence theorems for the $\overline{\partial } $ operator, Acta Math. 113 (1965), 89–152.
  • [H4] L. Hörmander, An Introduction to Complex Analysis in Several Variables, Van Nostrand, Princeton, 1966.
  • [HP] W. C. Hsiang and V. Pati, $L^{2}$-cohomology of normal algebraic surfaces, I, Invent. Math. 81 (1985), 395–412.
  • [KK] M. Kashiwara and T. Kawai, The Poincaré lemma for variations of polarized Hodge structure, Publ. Res. Inst. Math. Sci. 23 (1987), 345–407.
  • [K1] J. J. Kohn, Harmonic integrals on strongly pseudo-convex manifolds, I, Ann. of Math. (2) 78 (1963), 112–148.
  • [K2] J. J. Kohn, Harmonic integrals on strongly pseudo-convex manifolds, II, Ann. of Math. (2) 79 (1964), 450–472.
  • [KN] J. J. Kohn and L. Nirenberg, Non-coercive boundary value problems, Comm. Pure Appl. Math. 18 (1965), 443–492.
  • [K3] J. Kollár, “Singularities of pairs” in Algebraic Geometry (Santa Cruz 1995), Proc. Sympos. Pure Math. 62, Amer. Math. Soc., Providence, 1997, 221–287.
  • [LM] I. Lieb and J. Michel, The Cauchy-Riemann Complex, Vieweg, Braunschweig, Germany, 2002.
  • [M] R. MacPherson, “Global questions in the topology of singular spaces” in Proceedings of the International Congress of Mathematicians, Vol. 1 (Warsaw, 1983), PWN, Warsaw, 1984, 213–235.
  • [O] T. Ohsawa, Hodge spectral sequence on compact Kähler spaces, Publ. Res. Inst. Math. Sci. 23 (1987), 265–274.
  • [OV] N. Øvrelid and S. Vassiliadou, $L^{2}$-$\overline{\partial } $-cohomology groups of some singular complex spaces, Invent. Math. 192 (2013), 413–458.
  • [PS1] W. Pardon and M. Stern, $L^{2}$-$\overline{\partial } $-cohomology of complex projective varieties, J. Amer. Math. Soc. 4 (1991), 603–621.
  • [PS2] W. Pardon and M. Stern, Pure Hodge structure on the $L_{2}$-cohomology of varieties with isolated singularities, J. Reine Angew. Math. 533 (2001), 55–80.
  • [RR] J.-P. Ramis and G. Ruget, Complexe dualisant et théorème de dualité en géométrie analytique complexe, Inst. Hautes Études Sci. Publ. Math. 38 (1970), 77–91.
  • [R1] O. Riemenschneider, Characterizing Moišezon spaces by almost positive coherent analytic sheaves, Math. Z. 123 (1971), 263–284.
  • [R2] H. Rossi, Picard variety of an isolated singular point, Rice Univ. Studies 54 (1968), no. 4, 63–73.
  • [R3] J. Ruppenthal, About the $\overline{\partial } $-equation at isolated singularities with regular exceptional set, Internat. J. Math. 20 (2009), 459–489.
  • [T] K. Takegoshi, Relative vanishing theorems in analytic spaces, Duke Math. J. 52 (1985), 273–279.
  • [W] R. O. Wells, Differential Analysis on Complex Manifolds, 2nd ed., Grad. Texts in Math. 65, Springer, New York, 1980.