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Extending holomorphic forms from the regular locus of a complex space to a resolution of singularities

Authors: Stefan Kebekus and Christian Schnell
Journal: J. Amer. Math. Soc. 34 (2021), 315-368
MSC (2020): Primary 14B05, 14B15, 32S20
Published electronically: January 26, 2021
MathSciNet review: 4280862
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Abstract: We investigate under what conditions holomorphic forms defined on the regular locus of a reduced complex space extend to holomorphic (or logarithmic) forms on a resolution of singularities. We give a simple necessary and sufficient condition for this, whose proof relies on the Decomposition Theorem and Saito’s theory of mixed Hodge modules. We use it to generalize the theorem of Greb-Kebekus-Kovács-Peternell to complex spaces with rational singularities, and to prove the existence of a functorial pull-back for reflexive differentials on such spaces. We also use our methods to settle the “local vanishing conjecture” proposed by Mustaţă, Olano, and Popa.

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  • Carolina Araujo and Stéphane Druel, On Fano foliations, Adv. Math. 238 (2013), 70–118 DOI:10.1016/j.aim.2013.02.003. Preprint arXiv:1112.4512.
  • Carolina Araujo and Stéphane Druel, On codimension 1 del Pezzo foliations on varieties with mild singularities, Math. Ann. 360 (2014), no. 3–4, 769–798 DOI:10.1007/s00208-014-1053-3. Preprint arXiv:1210.4013.
  • Marco Andreatta and Alessandro Silva, On weakly rational singularities in complex analytic geometry, Ann. Mat. Pura Appl. (4) 136 (1984), 65–76 DOI:10.1007/BF01773377.
  • Daniel Barlet, The sheaf $\alpha ^\bullet _x$, J. Singul., 18 (2018), 50–83 DOI:10.5427/jsing.2018.18e. Preprint arXiv:1707.07962.
  • A. A. Beĭlinson, J. Bernstein, and P. Deligne, Faisceaux pervers, Analysis and topology on singular spaces, I (Luminy, 1981) Astérisque, vol. 100, Soc. Math. France, Paris, 1982, pp. 5–171 (French). MR 751966
  • Robert J. Berman and Henri Guenancia, Kähler-Einstein metrics on stable varieties and log canonical pairs, Geom. Funct. Anal. 24 (2014), no. 6, 1683–1730 DOI:10.1007/s00039-014-0301-8. Preprint arXiv:1304.2087.
  • Benjamin Bakker and Christian Lehn, The global moduli theory of symplectic varieties, Preprint arXiv:1812.09748, December 2018.
  • Edward Bierstone and Pierre D. Milman, Canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant, Invent. Math. 128 no. 2, 207–302 DOI:10.1007/s002220050141.
  • Bruno de Finetti, Theory of probability: a critical introductory treatment. Vol. 2, Wiley Series in Probability and Mathematical Statistics, John Wiley & Sons, London-New York-Sydney, 1975. Translated from the Italian by Antonio Machìand Adrian Smith. MR 0440641
  • Stéphane Druel, A decomposition theorem for singular spaces with trivial canonical class of dimension at most five, Invent. Math. 211 (2018), no. 1, 245–296 DOI:10.1007/s00222-017-0748-y. Preprint arXiv:1606.09006.
  • Hubert Flenner, Extendability of differential forms on nonisolated singularities, Invent. Math. 94 (1988), no. 2, 317–326. MR 958835, DOI 10.1007/BF01394328
  • Daniel Greb, Henri Guenancia, and Stefan Kebekus, Klt varieties with trivial canonical class: holonomy, differential forms, and fundamental groups, Geom. Topol. 23 (2019), no. 4, 2051–2124 DOI:10.2140/gt.2019.23.2051. Preprint arXiv:1704.01408.
  • Patrick Graf and Sándor J. Kovács, An optimal extension theorem for 1-forms and the Lipman-Zariski conjecture, Doc. Math. 19 (2014), 815–830. MR 3247804
  • Daniel Greb, Stefan Kebekus, and Sándor J. Kovács, Extension theorems for differential forms, and Bogomolov-Sommese vanishing on log canonical varieties, Compositio Math. 146 (2010), 193–219 DOI:10.1112/S0010437X09004321. A slightly extended version is available as arXiv:0808.3647.
  • Daniel Greb, Stefan Kebekus, Sándor J. Kovács, and Thomas Peternell, Differential forms on log canonical spaces, Inst. Hautes Études Sci. Publ. Math. 114 (2011), no. 1, 87–169 DOI:10.1007/s10240-011-0036-0. An extended version with additional graphics is available as arXiv:1003.2913.
  • Daniel Greb, Stefan Kebekus, and Thomas Peternell, Singular spaces with trivial canonical class, Minimal models and extremal rays (Kyoto, 2011) Adv. Stud. Pure Math., vol. 70, Math. Soc. Japan, [Tokyo], 2016, pp. 67–113. MR 3617779, DOI 10.2969/aspm/07010067
  • Daniel Greb, Stefan Kebekus, and Thomas Peternell, Étale fundamental groups of Kawamata log terminal spaces, flat sheaves, and quotients of abelian varieties, Duke Math. J. 165 (2016), no. 10, 1965–2004 DOI:10.1215/00127094-3450859. Preprint arXiv:1307.5718.
  • Daniel Greb, Stefan Kebekus, Thomas Peternell, and Behrouz Taji, Harmonic metrics on higgs sheaves and uniformisation of varieties of general type, Math. Ann., online first, 2019. DOI:10.1007/s00208-019-01906-4. Preprint arXiv:1804.01266.
  • Daniel Greb, Stefan Kebekus, Thomas Peternell, and Behrouz Taji, The Miyaoka-Yau inequality and uniformisation of canonical models, Ann. Sci. Éc. Norm. Supér. (4) 52 (2019), no. 6, 1487–1535 DOI:10.24033/asens.2414. Preprint arXiv:1511.08822.
  • Daniel Greb, Stefan Kebekus, Thomas Peternell, and Behrouz Taji, Nonabelian Hodge theory for klt spaces and descent theorems for vector bundles, Compos. Math. 155 (2019), no. 2, 289–323 DOI:10.1112/S0010437X18007923. Preprint arXiv:1711.08159.
  • Tommaso de Fernex, Brendan Hassett, Mircea Mustaţă, Martin Olsson, Mihnea Popa, and Richard Thomas (eds.), Algebraic geometry: Salt Lake City 2015, Proceedings of Symposia in Pure Mathematics, vol. 97, American Mathematical Society, Providence, RI; Clay Mathematics Institute, [Cambridge], MA, 2018. 2015 Summer Research Institute, Algebraic Geometry, July 13–31, 2015, University of Utah, Salt Lake City, Utah. MR 3793654, DOI 10.1090/pspum/097.1
  • Hans Grauert and Oswald Riemenschneider, Verschwindungssätze für analytische Kohomologiegruppen auf komplexen Räumen, Invent. Math. 11 (1970), 263–292 DOI:10.1007/BF01403182.
  • Hans Grauert and Reinhold Remmert, Coherent analytic sheaves, volume 265 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, 1984. DOI:10.1007/978-3-642-69582-7.
  • Phillip A. Griffiths, Variations on a theorem of Abel, Invent. Math. 35 (1976), 321–390. MR 435074, DOI 10.1007/BF01390145
  • A. Grothendieck, Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. III, Inst. Hautes Études Sci. Publ. Math. 28 (1966), 255. MR 217086
  • Robin Hartshorne, Algebraic geometry, Springer-Verlag, New York, 1977. Graduate Texts in Mathematics, No. 52. DOI:10.1007/978-1-4757-3849-0.
  • Annette Huber and Clemens Jörder, Differential forms in the h-topology, Algebr. Geom. 1 (2014), no. 4, 449–478 DOI:10.14231/AG-2014-020. Preprint arXiv:1305.7361.
  • Andreas Höring and Thomas Peternell, Algebraic integrability of foliations with numerically trivial canonical bundle, Invent. Math. 216 (2019), no. 2, 395–419 DOI:10.1007/s00222-018-00853-2. Preprint arXiv:1710.06183.
  • Ryoshi Hotta, Kiyoshi Takeuchi, and Toshiyuki Tanisaki, $D$-modules, perverse sheaves, and representation theory, Progress in Mathematics, vol. 236, Birkhäuser Boston, Inc., Boston, MA, 2008. Translated from the 1995 Japanese edition by Takeuchi. MR 2357361, DOI 10.1007/978-0-8176-4523-6
  • Annette Huber, Differential forms in algebraic geometry—a new perspective in the singular case, Port. Math. 73 (2016), no. 4, 337–367. MR 3580792, DOI 10.4171/PM/1990
  • Gavril Farkas and Ian Morrison (eds.), Handbook of moduli. Vol. II, Advanced Lectures in Mathematics (ALM), vol. 25, International Press, Somerville, MA; Higher Education Press, Beijing, 2013. MR 3183723
  • Stefan Kebekus, Pull-back morphisms for reflexive differential forms, Adv. Math. 245 (2013), 78–112 DOI:10.1016/j.aim.2013.06.013. Preprint arXiv:1210.3255.
  • János Kollár, Radu Laza, Giulia Saccà, and Claire Voisin, Remarks on degenerations of hyper-Kähler manifolds, Ann. Inst. Fourier (Grenoble) 68 (2018), no. 7, 2837–2882 (English, with English and French summaries). MR 3959097, DOI 10.5802/aif.3228
  • János Kollár and Shigefumi Mori, Birational geometry of algebraic varieties, volume 134 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, 1998. With the collaboration of C. H. Clemens and A. Corti, Translated from the 1998 Japanese original. DOI:10.1017/CBO9780511662560.
  • János Kollár, Singularities of the minimal model program, Cambridge Tracts in Mathematics, vol. 200, Cambridge University Press, Cambridge, 2013. With a collaboration of Sándor Kovács. MR 3057950, DOI 10.1017/CBO9781139547895
  • Sándor J. Kovács, Karl Schwede, and Karen E. Smith, The canonical sheaf of Du Bois singularities, Adv. Math. 224 (2010), no. 4, 1618–1640 DOI:10.1016/j.aim.2010.01.020.
  • Adrian Langer, Birational geometry of compactifications of Drinfeld half-spaces over a finite field, Adv. Math. 345 (2019), 861–908. MR 3902334, DOI 10.1016/j.aim.2019.01.031
  • Steven Lu and Behrouz Taji, A characterization of finite quotients of abelian varieties, Int. Math. Res. Not. IMRN (2018), no. 1, 292–319 DOI:10.1093/imrn/rnw251. Preprint arXiv:1410.0063.
  • Chi Li and Gang Tian, Orbifold regularity of weak Kähler-Einstein metrics, In Advances in complex geometry, volume 735 of Contemp. Math., pages 169–178. Amer. Math. Soc., Providence, RI, 2019. DOI:10.1090/conm/735/14825.
  • Mircea Mustaţă, Sebastián Olano, and Mihnea Popa, Local vanishing and Hodge filtration for rational singularities, J. Inst. Math. Jussieu 19 (2020), no. 3, 801–819 DOI:S1474748018000208. Preprint arXiv:1703.06704.
  • Yoshinori Namikawa, Deformation theory of singular symplectic $n$-folds, Math. Ann. 319 (2001), no. 3, 597–623 DOI:10.1007/PL00004451.
  • Th. Peternell, Modifications, Several complex variables, VII, Encyclopaedia Math. Sci., vol. 74, Springer, Berlin, 1994, pp. 285–317. MR 1326624, DOI 10.1007/978-3-662-09873-8_{8}
  • Mihnea Popa, Positivity for Hodge modules and geometric applications, Algebraic geometry: Salt Lake City 2015, Proc. Sympos. Pure Math., vol. 97, Amer. Math. Soc., Providence, RI, 2018, pp. 555–584. MR 3821162
  • Jean-Pierre Ramis, Gabriel Ruget, and Jean-Louis Verdier, Dualité relative en géométrie analytique complexe, Invent. Math. 13 (1971), 261–283 DOI:10.1007/BF01406078.
  • Morihiko Saito, Modules de Hodge polarisables, Publ. Res. Inst. Math. Sci. 24 (1988), no. 6, 849–995 (1989) (French). MR 1000123, DOI 10.2977/prims/1195173930
  • Morihiko Saito, Introduction to mixed Hodge modules, Astérisque 179-180 (1989), 10, 145–162. Actes du Colloque de Théorie de Hodge (Luminy, 1987). MR 1042805
  • Morihiko Saito, Mixed Hodge modules, Publ. Res. Inst. Math. Sci. 26 (1990), no. 2, 221–333. MR 1047415, DOI 10.2977/prims/1195171082
  • Morihiko Saito, On Kollár’s conjecture, Several complex variables and complex geometry, Part 2 (Santa Cruz, CA, 1989) Proc. Sympos. Pure Math., vol. 52, Amer. Math. Soc., Providence, RI, 1991, pp. 509–517. MR 1128566
  • Morihiko Saito, On the theory of mixed Hodge modules [ MR1150484 (93e:14014)], Selected papers on number theory, algebraic geometry, and differential geometry, Amer. Math. Soc. Transl. Ser. 2, vol. 160, Amer. Math. Soc., Providence, RI, 1994, pp. 47–61. MR 1308540, DOI 10.1090/trans2/160/04
  • Christian Schnell, An overview of Morihiko Saito’s theory of mixed Hodge modules, Preprint arXiv:1405.3096, May 2014.
  • Christian Schnell, On Saito’s vanishing theorem, Math. Res. Lett. 23 (2016), no. 2, 499–527. MR 3512896, DOI 10.4310/MRL.2016.v23.n2.a10
  • A. Seidenberg, The hyperplane sections of normal varieties, Trans. Amer. Math. Soc. 69 (1950), 357–386. MR 37548, DOI 10.1090/S0002-9947-1950-0037548-0
  • The Stacks Project Authors, Stacks project, Available on the internet at, 2018.
  • Duko van Straten and Joseph Steenbrink, Extendability of holomorphic differential forms near isolated hypersurface singularities, Abh. Math. Sem. Univ. Hamburg 55 (1985), 97–110 DOI:10.1007/BF02941491.

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Additional Information

Stefan Kebekus
Affiliation: Mathematisches Institut, Albert-Ludwigs-Universität Freiburg, Ernst-Zermelo- Straße 1, 79104 Freiburg im Breisgau, Germany; and Freiburg Institute for Advanced Studies (FRIAS), Freiburg im Breisgau, Germany
MR Author ID: 637173

Christian Schnell
Affiliation: Department of Mathematics, Stony Brook University, Stony Brook, Nnew York 11794-3651
MR Author ID: 904612

Keywords: Extension theorem, holomorphic form, complex space, mixed Hodge module, decomposition theorem, rational singularities, pull-back.
Received by editor(s): February 19, 2019
Received by editor(s) in revised form: January 21, 2020
Published electronically: January 26, 2021
Additional Notes: The first author was supported by a senior fellowship of the Freiburg Institute of Advanced Studies (FRIAS)
During the preparation of this paper, the second author was supported by a Mercator Fellowship from the Deutsche Forschungsgemeinschaft (DFG), by research grant DMS-1404947 from the National Science Foundation (NSF), and by the Kavli Institute for the Physics and Mathematics of the Universe (IPMU) through the World Premier International Research Center Initiative (WPI initiative), MEXT, Japan.
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