Jean Ruppenthal

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.

L 2 -RIEMANN-ROCH FOR SINGULAR COMPLEX CURVES

We present a comprehensive L 2 -theory for the @-operator on singular complex curves, including L 2 -versions of the Riemann-Roch theorem and some applications.

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.

The Dolbeault complex with weights according to normal crossings

In the present paper, we define a Dolbeault complex with weights according to normal crossings, which is a useful tool for studying the

THE ∂-EQUATION ON HOMOGENEOUS VARIETIES WITH AN ISOLATED SINGULARITY

{{\bar{\partial}}}

Compactness of the ∂¯-Neumann operator on singular complex spaces

-equation on singular complex spaces by resolution of singularities (where normal crossings appear naturally). The major difficulty is to prove that this complex is locally exact. We do that by constructing a local