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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
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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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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

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

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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