Richard Lärkäng

On the duality theorem on an analytic variety

The duality theorem for Coleff-Herrera products on a complex manifold says that if f = (f1, . . . , fp) defines a complete intersection, then the annihilator of the Coleff-Herrera product μ equals (locally) the ideal generated by f . This does not hold unrestrictedly on an analytic variety Z. We give necessary, and in many cases sufficient conditions for when the duality theorem holds. These conditions are related to how the zero set of f intersects certain singularity subvarieties of the sheaf OZ .

VARIOUS APPROACHES TO PRODUCTS OF RESIDUE CURRENTS

We describe various approaches to Coleff-Herrera products of residue currents R (of Cauchy-Fantappiè-Leray type) associated to holomorphic mappings fj. More precisely, we study to which extent (exterior) products of natural regularizations of the individual currents R yield regularizations of the corresponding Coleff-Herrera products. Our results hold globally on an arbitrary pure-dimensional complex space.

Residue currents associated with weakly holomorphic functions

We construct Coleff–Herrera products and Bochner–Martinelli type residue currents associated with a tuple f of weakly holomorphic functions, and show that these currents satisfy basic properties from the (strongly) holomorphic case. This include the transformation law, the Poincaré–Lelong formula and the equivalence of the Coleff–Herrera product and the Bochner– Martinelli type residue current associated with f when f defines a complete intersection.

Computing residue currents of monomial ideals using comparison formulas

Given a free resolution of an ideal a of holomorphic functions, one can construct a vector-valued residue current R, which coincides with the classical Coleff–Herrera product if a is a complete intersection ideal and whose annihilator ideal is precisely a. We give a complete description of R in the case when a is an Artinian monomial ideal and the resolution is the hull resolution (or a more general cellular resolution). The main ingredient in the proof is a comparison formula for residue currents due to the first author. By means of this description, we obtain in the monomial case a current version of a factorization of the fundamental cycle of a due to Lejeune-Jalabert.

Koppelman Formulas on Affine Cones Over Smooth Projective Complete Intersections

In the present paper, we study regularity of the Andersson-Samuelsson Koppelman integral operator on affine cones over smooth projective complete intersections. Particularly, we prove L-p- and C-alpha-estimates, and compactness of the operator, when the degree is sufficiently small. As applications, we obtain homotopy formulas for different partial derivative-operators acting on L-p-spaces of forms, including the case p = 2 if the varieties have canonical singularities. We also prove that the A-forms introduced by Andersson-Samuelsson are C-alpha for alpha < 1.

Koppelman formulas on the A1-singularity

In the present paper, we study the regularity of the Andersson-Samuelsson Koppelman integral operator on the A_1-singularity. Particularly, we prove L^p- and C^0-estimates. As applications, we obtain L^p-homotopy formulas for the dbar-equation on the A_1-singularity, and we prove that the A-forms introduced by Andersson-Samuelsson are continuous on the A_1-singularity.

Residue currents with prescribed annihilator ideals on singular varieties

Given an ideal J on a complex manifold, Andersson and Wulcan constructed a current R such that the annihilator of R is J , generalizing the duality theorem for Coleff-Herrera products. We describe a way to generalize this construction to ideals on singular varieties.

Chern forms of singular metrics on vector bundles

We study singular hermitian metrics on holomorphic vector bundles, following Berndtsson-Păun. 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.

Residue Currents and Fundamental Cycles

We give a factorization of the fundamental cycle of an analytic space in terms of certain differential forms and residue currents associated with a locally free resolution of its structure sheaf. Our result can be seen as a generalization of the classical Poincare-Lelong formula. It is also a current version of a result by Lejeune-Jalabert, who similarly expressed the fundamental class of a Cohen-Macaulay analytic space in terms of differential forms and cohomological residues.

Extending holomorphic maps from Stein manifolds into affine toric varieties

A complex manifold Y is said to have the interpolation property if a holomorphic map to Y from a subvariety S of a reduced Stein space X has a holomorphic extension to X if it has a continuous extension. Taking S to be a contractible submanifold of X = C^n gives an ostensibly much weaker property called the convex interpolation property. By a deep theorem of Forstneric, the two properties are equivalent. They (and about a dozen other nontrivially equivalent properties) define the class of Oka manifolds. This paper is the first attempt to develop Oka theory for singular targets. The targets that we study are affine toric varieties, not necessarily normal. We prove that every affine toric variety satisfies a weakening of the interpolation property that is much stronger than the convex interpolation property, but the full interpolation property fails for most affine toric varieties, even for a source as simple as the product of two annuli embedded in C^4.