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

By Stefan Kebekus, C. Schnell. Journal of The American Mathematical Society, 2018

89 citations recorded by the source corpus.

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.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/64b6b6e45a62ba5c390a3b4af598cd7260bfbc01

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In Geometry and Topology

1. Mihnea Cristian Popa
2. M. Mustaţă
3. C. Lehn
4. Benjamin Bakker
5. Chi Li
6. S. Ishii
7. H. Blum
8. Mattias Jonsson

In Mathematical Physics

1. M. Mustaţă
2. Mihnea Cristian Popa
3. C. Lehn
4. Benjamin Bakker
5. S. Ishii
6. Tommaso de Fernex
7. C. Schnell
8. L. Ein

Citations in the corpus

• The global moduli theory of symplectic varieties (2018, result): “(i) An analysis of the Hodge theory of rational and symplectic singularities in the non-projective setting, using recent results of Kebekus–Schnell [KS18] on extending holomorphic forms.”
• Hodge filtration on local cohomology, Du Bois complex and local cohomological dimension (2021, methodology): “When Z has isolated singularities, we refine a result in [25] on the coincidence of h-differentials and reflexive differentials, for forms of low degree.”