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

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.

arXiv commentFinal version. To appear in slightly shortened version in J. Amer. Math. Soc

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• Differential Forms on Log Canonical Spaces
• Étale fundamental groups of Kawamata log terminal spaces, flat sheaves, and quotients of Abelian varieties
• The canonical sheaf of Du Bois singularities
• Klt varieties with trivial canonical class -- Holonomy, differential forms, and fundamental groups
• The Miyaoka-Yau inequality and uniformisation of canonical models
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• Nonabelian Hodge Theory for klt spaces and descent theorems for vector bundles
• Nakamaye's theorem on complex manifolds
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Cited by in corpus (7)

• The global moduli theory of symplectic varieties
• Projectively flat KLT varieties
• Differential forms on log canonical spaces in positive characteristic
• Kähler spaces with zero first Chern class: Bochner principle, Albanese map and fundamental groups
• The Kodaira problem for Kähler spaces with vanishing first Chern class
• Weighted multiplier ideals of reduced divisors
• A decomposition theorem for singular Kähler spaces with trivial first Chern class of dimension at most four