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Jean Ruppenthal

Publications and source records attributed to Jean Ruppenthal.

27 records · Page 2Linked to original sources
arXiv2010

Compactness of the d-bar-Neumann operator on singular complex spaces

Let X be a Hermitian complex space of pure dimension n. We show that the d-bar-Neumann operator on (p,q)-forms is compact at isolated singularities of X if q>0 and p+q is not equal to n-1 or n. The main step is the construction of compact solution operators for the d-bar-equation on such spaces which is based on a general characterization of compactness in function spaces on singular spaces, and that leads also to a criterion for compactness of more general Green operators on singular spaces.

math.CV↗
arXiv2009

Friedrichs' extension lemma with boundary values and applications in complex analysis

Let $Q$ be a first-order differential operator on a compact, smooth oriented Riemannian manifold with smooth boundary. Then, Friedrichs' extension lemma states that the minimal closed extension $Q_{min}$ (the closure of the graph) and the maximal closed extension $Q_{max}$ (in the sense of distributions) of $Q$ in $L^p$-spaces ($1\leq p<\infty$) coincide. In the present paper, we show that the same is true for boundary values with respect to $Q_{min}$ and $Q_{max}$. This gives a useful characterization of weak boundary values, particularly for $Q=d-bar$ the Cauchy-Riemann operator. As an application, we derive the Bochner-Martinelli-Koppelman formula for $L^p$-forms with weak d-bar-boundary values.

math.AP↗
arXiv2009

About the d-bar-equation at isolated singularities with regular exceptional set

Let Y be a pure dimensional analytic variety in C^n with an isolated singularity at the origin such that the exceptional set X of a desingularization of Y is regular. The main objective of this paper is to present a technique which allows to determine obstructions to the solvability of the d-bar-equation in the $L^2$ respectively $L^\infty$ sense on Y^*=Y-{0} in terms of certain cohomology classes on X. More precisely, let D be a Stein domain, relatively compact in Y, containing the origin, D^*=D-{0}. We give a sufficient condition for the solvability of the d-bar-equation in the $L^2$-sense on D^*; and in the $L^\infty$ sense, if D is in addition strongly pseudoconvex. If Y is an irreducible cone, we also give some necessary conditions and obtain optimal Hoelder estimates for solutions of the d-bar-equation.

math.CV↗
arXiv2009

The Dolbeault complex with weights according to normal crossings

In this paper, we define a Dolbeault complex with weights according to normal crossings, which is a useful tool for studying the d-bar-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 d-bar-solution operator which involves only Cauchy's Integral Formula (in one complex variable) and behaves well for L^p-forms with weights according to normal crossings.

math.CV↗
arXiv2009

The d-bar-equation on homogeneous varieties with an isolated singularity

Let X be a regular irreducible variety in CP^{n-1}, Y the associated homogeneous variety in C^n, and N the restriction of the universal bundle of CP^{n-1} to X. In the present paper, we compute the obstructions to solving the d-bar-equation in the L^p-sense on Y for 1<=p<=\infty in terms of cohomology groups H^q(X,O(N^m)). That allows to identify obstructions explicitly if X is specified more precisely, for example if it is equivalent to CP^1 or an elliptic curve.

math.CV↗
arXiv2009

A d-bar-theoretical proof of Hartogs' extension theorem on (n-1)-complete spaces

Let X be a connected normal complex space of dimension n>=2 which is (n-1)-complete, and let p: M -> X be a resolution of singularities. By use of Takegoshi's generalization of the Grauert-Riemenschneider vanishing theorem, we deduce H^1_{cpt}(M,O)=0, which in turn implies Hartogs' extension theorem on X by the d-bar-technique of Ehrenpreis.

math.CV↗
arXiv2008

An explicit d-bar-integration formula for weighted homogeneous varieties II: forms of higher degree

Let Y be a weighted homogeneous (singular) subvariety of C^n. The main objective of this paper is to present a class of explicit integral formulae for solving the d-bar-equation $ω=\dbarλ$ on the regular part of Y, where $ω$ is a d-bar-closed (0,q)-form with compact support and degree q>=1. Particular cases of these formulae yield L^p-bounded solution operators for $1<=p<=\infty$ if Y is a homogeneous and pure dimensional subvariety with an arbitrary singular locus.

math.CV↗
arXiv2008

An explicit d-bar-integration formula for weighted homogeneous varieties

Let Y be a weighted homogeneous (singular) subvariety of C^n. The main objective of this paper is to present an explicit formula for solving the d-bar-equation $f=\dbar{g}$ on the regular part of Y, where $f$ is a d-bar-closed $(0,1)$-form with compact support. This formula will then be used to give Hölder estimates for the solution in case $Y$ 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 Y is pure dimensional and homogeneous.

math.CV↗