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.

Final version. To appear in slightly shortened version in J. Amer. Math. Soc

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• 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