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Journal of the American Mathematical Society

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ISSN 1088-6834 (online) ISSN 0894-0347 (print)

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Extending holomorphic forms from the regular locus of a complex space to a resolution of singularities
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by Stefan Kebekus and Christian Schnell
J. Amer. Math. Soc. 34 (2021), 315-368
Published electronically: January 26, 2021


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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Bibliographic 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
  • Email:
  • Christian Schnell
  • Affiliation: Department of Mathematics, Stony Brook University, Stony Brook, Nnew York 11794-3651
  • MR Author ID: 904612
  • Email:
  • 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.
  • © Copyright 2021 American Mathematical Society
  • Journal: J. Amer. Math. Soc. 34 (2021), 315-368
  • MSC (2020): Primary 14B05, 14B15, 32S20
  • DOI:
  • MathSciNet review: 4280862