A Remark on δ-cohomology with supports in the complement of a cone singularity

Abstract

Let [this equation] be a ∂-closed form on ℂⁿ x ℂ. If it is assumed that ψ has compact (specifically non-empty) support, then ψn+1(w,z) ≠ 0, for if the last component vanishes identically then this, together with the closedness of ψ, implies that [this second equation] vanishes identically, 1 ≤ k ≤ n. Hence ψ is the pullback of a form defined on ℂⁿ, which contradicts the compactness of support.