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Jean Ruppenthal

Publications and source records attributed to Jean Ruppenthal.

At least 19 recordsLinked to original sources
arXiv2026

Canonical sheaves at isolated canonical Gorenstein singularities

It is well known that the Grauert-Riemenschneider canonical sheaf $\mathcal{K}_X$ of holomorphic square-integrable $n$-forms is a central tool in $L^2$-theory for the $\overline\partial$-operator on a singular complex space $X$ of pure dimension $n$. It was shown a few years ago that a comprehensive $L^2$-theory requires also the study of the sheaf $\mathcal{K}_X^s$ of holomorphic square-integrable $n$-forms with a Dirichlet boundary condition at the singular set of $X$. In the present paper, we describe and classify the behaviour of $\mathcal{K}_X^s$ in isolated canonical Gorenstein singularities, and give applications to the $L^2$-theory for the $\overline\partial$-operator on such spaces.

math.CV↗
arXiv2018

Dolbeault cohomology of complex nilmanifolds foliated in toroidal groups

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.

math.DG↗
arXiv2018

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 $C^α$-estimates, and compactness of the operator, when the degree is sufficiently small. As applications, we obtain homotopy formulas for different $\overline{\partial}$-operators acting on $L^p$-spaces of forms, including the case $p=2$ if the varieties have canonical singularities. We also prove that the $\mathcal{A}$-forms introduced by Andersson-Samuelsson are $C^α$ for $α< 1$.

math.CV↗
arXiv2018

Estimates for the $\bar{\partial}$-equation on canonical surfaces

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.

math.CV↗
arXiv2017

Chern forms of singular metrics on vector bundles

We study singular hermitian metrics on holomorphic vector bundles, following Berndtsson-P{ă}un. 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.

math.CV↗
arXiv2017

Koppelman formulas on the A_1-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 $\bar{\partial}$-equation on the $A_1$-singularity, and we prove that the $\mathcal{A}$-forms introduced by Andersson-Samuelsson are continuous on the $A_1$-singularity.

math.CV↗
arXiv2016

Explicit Serre duality on complex spaces

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.

math.CV↗
arXiv2016

Modifications of torsion-free coherent analytic sheaves

We study the transformation of torsion-free coherent analytic sheaves under proper modifications. More precisely, we study direct images of inverse image sheaves, and torsion-free preimages of direct image sheaves. Under some conditions, it is shown that torsion-free coherent sheaves can be realized as the direct image of locally free sheaves under modifications. Thus, it is possible to study coherent sheaves modulo torsion by reducing the problem to study vector bundles on manifolds. We apply this to reduced ideal sheaves and to the Grauert-Riemenschneider canonical sheaf of holomorphic n-forms.

math.CV↗
arXiv2016

$L^2$-Serre duality on singular complex spaces and rational singularities

In the present paper, we devise a version of topological $L^2$-Serre duality for singular complex spaces with arbitrary singularities. This duality is used to deduce various new $L^2$-vanishing theorems for the $\bar{\partial}$-equation on singular spaces. It is shown that complex spaces with rational singularities behave quite tame with respect to the $\bar{\partial}$-equation in the $L^2$-sense. More precisely: a singular point is rational if and only if the $L^2$-$\bar{\partial}$-complex is exact in this point. So, we obtain an $L^2$-$\bar{\partial}$-resolution of the structure sheaf in rational singular points.

math.CV↗
arXiv2015

$L^2$-Riemann-Roch for singular complex curves

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

math.CV↗
arXiv2015

Parabolicity of the regular locus of complex varieties

The purpose of this note is to show that the regular locus of a complex variety is locally parabolic at the singular set. This yields that the regular locus of a compact complex variety, e.g., of a projective variety, is parabolic. We give also an application to the $L^2$-theory for the $\overline{\partial}$-operator on singular spaces.

math.CV↗
arXiv2015

$L^2$-sheaves of holomorphic functions and $n$-forms on complex spaces with isolated singularities

Let $X$ be a Hermitian complex space of pure dimension $n$ with isolated singularities. In the present paper, we give a natural resolution for the canonical sheaf of square-integrable holomorphic $n$-forms with Dirichlet boundary condition on $X$. As application, we obtain an explicit smooth model for the $L^2$-$\bar{\partial}$-cohomology, including natural resolutions for sheaves of $\bar{\partial}$-closed (holomorphic) $L^2$-functions.

math.CV↗
arXiv2014

$L^2$-Serre duality on singular complex spaces and applications

In this survey, we explain a version of topological $L^2$-Serre duality for singular complex spaces with arbitrary singularities. This duality can be used to deduce various $L^2$-vanishing theorems for the $\overline\partial$-equation on singular spaces. As one application, we prove Hartogs' extension theorem for $(n-1)$-complete spaces. Another application is the characterization of rational singularities. It is shown that complex spaces with rational singularities behave quite tame with respect to some $\overline\partial$-equation in the $L^2$-sense. More precisely: a singular point is rational if and only if the appropriate $L^2$-$\overline\partial$-complex is exact in this point. So, we obtain an $L^2$-$\overline\partial$-resolution of the structure sheaf in rational singular points.

math.CV↗
arXiv2014

$L^2$-theory for the $\overline\partial$-operator on compact complex spaces

Let $X$ be a singular Hermitian complex space of pure dimension $n$. We use a resolution of singularities to give a smooth representation of the $L^2$-$\overline\partial$-cohomology of $(n,q)$-forms on $X$. The central tool is an $L^2$-resolution for the Grauert-Riemenschneider canonical sheaf $\mathcal{K}_X$. As an application, we obtain a Grauert-Riemenschneider-type vanishing theorem for forms with values in almost positive line bundles. If $X$ is a Gorenstein space with canonical singularities, then we get also an $L^2$-representation of the flabby cohomology of the structure sheaf $\mathcal{O}_X$. To understand also the $L^2$-$\overline\partial$-cohomology of $(0,q)$-forms on $X$, we introduce a new kind of canonical sheaf, namely the canonical sheaf of square-integrable holomorphic $n$-forms with some (Dirichlet) boundary condition at the singular set of $X$. If $X$ has only isolated singularities, then we use an $L^2$-resolution for that sheaf and a resolution of singularities to give a smooth representation of the $L^2$-$\overline\partial$-cohomology of $(0,q)$-forms.

math.CV↗
arXiv2013

Adjunction for the Grauert-Riemenschneider canonical sheaf and extension of L2-cohomology classes

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.

math.CV↗
arXiv2012

L2-Properties of the dbar and the dbar-Neumann operator on spaces with isolated singularities

Let X be a Hermitian complex space of pure dimension with only isolated singularities and p: M -> X a resolution of singularities. Let D be a relatively compact domain in X with no singularities in the boundary, D^*=D-Sing(X) the regular part of D and D'=p^{-1}(D) the preimage of D under p. We relate L2-properties of the dbar and the dbar-Neumann operator on D^* to properties of the corresponding operators on D' (where the situation is classically well understood). Outside some middle degrees, there are compact solution operators for the dbar-equation on D^* exactly if there are such operators on the resolution D', and the dbar-Neumann operator is compact on D^* exactly if it is compact on D'.

math.CV↗