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

Kebekus, Stefan; Schnell, Christian

doi:10.1090/jams/962

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

DOI: https://doi.org/10.1090/jams/962

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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