Koppelman Formulas and Existence Theorems for the ∂̄-equation on Analytic Varieties

Let Z be an analytic subvariety of pure codimension p of a pseudoconvex set X in C. We introduce weighted Koppelman formulas on Z that provides solutions to the ∂̄-equation. If φ is a smooth (0, q + 1)-form on Z with ∂̄φ = 0, we find a smooth (0, q)-form ψ on Zreg with at most polynomial growth at Zsing such that ∂̄ψ = φ. We also present Hartogs theorems for ∂̄-closed forms on Z. As a consequence we find that the Dolbeault cohomology H(Zreg) vanishes if q < ν−dimZsing −1, where ν is the (minimal) depth of the stalks of the structure sheaf ØZ = Ø/J of Z.