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We solve the $${\bar{\partial }}$$ ∂ ¯ -equation for (p, q)-forms locally on any reduced pure-dimensional complex space and we prove an explicit version of Serre duality by introducing suitable concrete fine sheaves of certain (p, q)-currents. In particular this gives a condition for the $${\bar{\partial }}$$ ∂ ¯ -equation to be globally solvable. Our results also give information about holomorphic p-forms on singular spaces.

We study the solvability in $L^p$ of the $\bar\partial$-equation in a neighborhood of a canonical singularity on a complex surface, a so-called du Val singularity. We get a quite complete picture in case $p=2$ for two natural closed extensions $\bar\partial_s$ and $\bar\partial_w$ of $\bar\partial$. For $\bar\partial_s$ we have solvability, whereas for $\bar\partial_w$ there is solvability if and only if a certain boundary condition $(*)$ is fulfilled at the singularity. Our main tool is certain integral operators for solving $\bar\partial$ introduced by the first and fourth author, and we study mapping properties of these operators at the singularity., 21 pages

We develop intersection theory in terms of the $${{\mathscr {B}}}$$ B -group of a reduced analytic space. This group was introduced in a previous work as an analogue of the Chow group; it is generated by currents that are direct images of Chern forms and it contains all usual cycles. However, contrary to Chow classes, the $${{\mathscr {B}}}$$ B -classes have well-defined multiplicities at each point. We focus on a $${{\mathscr {B}}}$$ B -analogue of the intersection theory based on the Stückrad–Vogel procedure and the join construction in projective space. Our approach provides global $${{\mathscr {B}}}$$ B -classes which satisfy a Bézout theorem and have the expected local intersection numbers. We also introduce $${{\mathscr {B}}}$$ B -analogues of more classical constructions of intersections using the Gysin map of the diagonal. These constructions are connected via a $${{\mathscr {B}}}$$ B -variant of van Gastel’s formulas. Furthermore, we prove that our intersections coincide with the classical ones on cohomology level.

Given equidimensional (generalized) cycles $$\mu _1$$ μ 1 and $$\mu _2$$ μ 2 on a complex manifold Y we introduce a product $$\mu _1\diamond _{Y} \mu _2$$ μ 1 ⋄ Y μ 2 that is a generalized cycle whose multiplicities at each point are the local intersection numbers at the point. If Y is projective, then given a very ample line bundle $$L\rightarrow Y$$ L → Y we define a product $$\mu _1{\bullet _L}\mu _2$$ μ 1 ∙ L μ 2 whose multiplicities at each point also coincide with the local intersection numbers. In addition, provided that $$\mu _1$$ μ 1 and $$\mu _2$$ μ 2 are effective, this product satisfies a Bézout inequality. If $$i:Y\rightarrow {\mathbb P}^N$$ i : Y → P N is an embedding such that $$i^*\mathcal O(1)=L$$ i ∗ O ( 1 ) = L , then $$\mu _1{\bullet _L}\mu _2$$ μ 1 ∙ L μ 2 can be expressed as a mean value of Stückrad–Vogel cycles on $${\mathbb P}^N$$ P N . There are quite explicit relations between $${\diamond }_Y$$ ⋄ Y and $${\bullet _L}$$ ∙ L .

On a reduced analytic space X we introduce the concept of a generalized cycle, which extends the notion of a formal sum of analytic subspaces to include also a form part. We then consider a suitable equivalence relation and corresponding quotient $$\mathcal {B}(X)$$ B ( X ) that we think of as an analogue of the Chow group and a refinement of de Rham cohomology. This group allows us to study both global and local intersection theoretic properties. We provide many $$\mathcal {B}$$ B -analogues of classical intersection theoretic constructions: For an analytic subspace $$V\subset X$$ V ⊂ X we define a $$\mathcal {B}$$ B -Segre class, which is an element of $$\mathcal {B}(X)$$ B ( X ) with support in V. It satisfies a global King formula and, in particular, its multiplicities at each point coincide with the Segre numbers of V. When V is cut out by a section of a vector bundle we interpret this class as a Monge–Ampère-type product. For regular embeddings we construct a $$\mathcal {B}$$ B -analogue of the Gysin morphism.

We make the classical Dickenstein-Sessa canonical representation in local moderate cohomology explicit by an integral formula. We also provide a similar representation of the higher local moderate cohomology groups. The results are related to holomorphic forms on non-reduced complex spaces., Comment: revised version, 15 pages

We show that Coleff–Herrera type products of residue currents can be defined by analytic continuation of natural functions depending on one complex variable.

The Ronkin function plays a fundamental role in the theory of amoebas. We introduce an analogue of the Ronkin function in the setting of coamoebas. It turns out to be closely related to a certain toric arrangement known as the shell of the coamoeba and we use our Ronkin type function to obtain some properties of it., Comment: 2 figures. Comments are welcome!

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 concrete fine sheaves $\mathscr{B}_X^{n,q}$ of certain currents on $X$ of bidegree $(n,q)$, such that the Dolbeault complex $(\mathscr{B}_X^{n,\bullet},\,\bar{\partial})$ becomes, in a certain sense, a dualizing complex. In particular, if $X$ is Cohen-Macaulay (e.g., Gorenstein or a complete intersection) then $(\mathscr{B}_X^{n,\bullet},\,\bar{\partial})$ is an explicit fine resolution of the Grothendieck dualizing sheaf., Final version

We study extensions of Wermer's maximality theorem to several complex variables. We exhibit various smoothly embedded manifolds in complex Euclidean space whose hulls are non-trivial but contain no analytic disks. We answer a question posed by Lee Stout concerning the existence of analytic structure for a uniform algebra whose maximal ideal space is a manifold., Comment: Comments are welcome!

In the present paper, we derive an adjunction formula for the Grauert-Riemenschneider canonical sheaf of a singular hypersurface V in a complex manifold M. This adjunction formula is used to study the problem of extending L2-cohomology classes of dbar-closed forms from the singular hypersurface V to the manifold M in the spirit of the Ohsawa-Takegoshi-Manivel extension theorem. We do that by showing that our formulation of the L2-extension problem is invariant under bimeromorphic modifications, so that we can reduce the problem to the smooth case by use of an embedded resolution of V in M. The smooth case has recently been studied by Berndtsson., Comment: 20 pages

We describe various approaches to Coleff-Herrera products of residue currents $R^j$ (of Cauchy-Fantappi\`e-Leray type) associated to holomorphic mappings $f_j$. More precisely, we study to which extent (exterior) products of natural regularizations of the individual currents $R^j$ yield regularizations of the corresponding Coleff-Herrera products. Our results hold globally on an arbitrary pure-dimensional complex space., Comment: 23 pages, rewritten introduction