Let X be a Hermitian complex space of pure dimension with only isolated singularities and π : M → X a resolution of singularities. Let Ω ⊂⊂ X be a domain with no singularities in the boundary, Ω * = Ω \ Sing X and Ω ′ = π −1 (Ω). We relate L 2 -properties of the ∂ and the ∂-Neumann operator on Ω * to properties of the corresponding operators on Ω ′ (where the situation is classically well understood). Outside some middle degrees, there are compact solution operators for the ∂-equation on Ω * exactly if there are such operators on the resolution Ω ′ , and the ∂-Neumann operator is compact on Ω * exactly if it is compact on Ω ′ .

In this paper we use recently developed calculus of residue currents together with integral formulas to give a new explicit analytic realization, as well as a new analytic proof, of Serre duality on any reduced pure n-dimensional paracompact complex space X. At the core of the paper is the introduction of certain fine sheaves B n,q X of currents on X of bidegree (n, q), such that the Dolbeault complex (B n,• X ,∂) becomes, in a certain sense, a dualizing complex. In particular, if X is Cohen-Macaulay then (B n,• X ,∂) is an explicit fine resolution of the Grothendieck dualizing sheaf. c (X, F * ⊗ Ω n X ) is replaced by Ext −q c (X; F , K • ), where K • is the dualizing complex in the sense of Ramis-Ruget [27], that is a certain complex of O X -modules with coherent cohomology.

Let Σ be a weighted homogeneous (singular) subvariety of C n . The main objective of this paper is to present an explicit formula for solving the ∂-equation λ = ∂g on the regular part of Σ, where λ is a ∂-closed (0, 1)-form with compact support. This formula will then be used to give Hölder estimates for the solution in case Σ is homogeneous (a cone) with an isolated singularity. Finally, a slight modification of our formula also gives an L 2 -bounded solution operator in case Σ is pure d-dimensional and homogeneous.

Abstract. We study singular hermitian metrics on holomorphic vector bundles, following Berndtsson-Pȃun. Previous work by Raufi has shown that for such metrics, it is in general not possible to define the curvature as a current with measure coefficients. In this paper we show that despite this, under appropriate codimension restrictions on the singular set of the metric, it is still possible to define Chern forms as closed currents of order 0 with locally finite mass, which represent the Chern classes of the vector bundle.

It is conjectured that the Dolbeault cohomology of a complex nilmanifold X is computed by left-invariant forms. We prove this under the assumption that X is suitably foliated in toroidal groups and deduce that the conjecture holds in real dimension up to six.Our approach generalises previous methods, where the existence of a holomorphic fibration was a crucial ingredient.