Related papers: $L^2$-sheaves of holomorphic functions and $n$-for…

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…

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…

We present a refined, improved $L^2$-theory for the $\bar{\partial}$-operator for $(0,q)$ and $(n,q)$-forms on Hermitian complex spaces of pure dimension $n$ with isolated singularities. The general philosophy is to use a resolution of…

We construct a generalization of the Dolbeault-Grothendieck resolution on a singular complex space. The same construction yields, for each morphism of analytic spaces, a pullback mapping between the respective Dolbeault-Grothendieck…

For a reduced pure dimensional complex space $X$, we show that if Barlet's recently introduced sheaf $\alpha_X^1$ of holomorphic $1$-forms or the sheaf of germs of weakly holomorphic $1$-forms is locally free, then $X$ is smooth. Moreover,…

The main problem studied is resolution of singularities of the cotangent sheaf of a complex- or real-analytic variety Y (or of an algebraic variety Y over a field of characteristic zero). Given Y, we ask whether there is a global resolution…

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…

Let $M$ be an analytic manifold over $\mathbb{R}$ or $\mathbb{C}$, $\theta$ a $1$-dimensional Log-Canonical (resp. monomial) singular distribution and $\mathcal{I}$ a coherent ideal sheaf defined on $M$. We prove the existence of a…

Dosi and, quite recently, the author showed that, on the character space of a nilpotent Lie algebra, there exists a sheaf of Fr\'echet--Arens--Michael algebras (of noncommutative holomorphic functions in the complex case and of…

Let X be a pure n-dimensional (n>1) complex analytic set in C^N with an isolated singularity at 0. In this paper we express the L2-(0,q)-d-bar-cohomology groups for all q with 0<q<n+1, of a sufficiently small deleted neighborhood of the…

In this paper, in order to develop a more general $L^2$-theory for the $\overline{\partial}$-operator on complex spaces, we provide $L^2$-Dolbeault fine resolutions and isomorphisms, and $L^2$-estimates, for holomorphic line bundles on…

In this article, we first establish an $L^2$-type Dolbeault isomorphism for the sheaf of logarithmic differential forms twisted by the multiplier ideal sheaf. By using this isomorphism and $L^2$-estimates equipped with a singular Hermitian…

Let $X$ be a complex space of pure dimension. We introduce fine sheaves $\A^X_q$ of $(0,q)$-currents, which coincides with the sheaves of smooth forms on the regular part of $X$, so that the associated Dolbeault complex yields a resolution…

We introduce smooth L^\infty differential forms on a singular (semialgebraic) set X in R^n. Roughly speaking, a smooth L^\infty differential form is a certain class of equivalence of 'stratified forms', that is, a collection of smooth forms…

Given a complex manifold containing a relatively compact $Z(q)$ domain, we give sufficient geometric conditions on the domain so that its $L^2$-cohomology in degree $(p,q)$ (known to be finite dimensional) vanishes. The condition consists…

On any pure $n$-dimensional, possibly non-reduced, analytic space $X$ we introduce the sheaves $\mathscr{E}_X^{p,q}$ of smooth $(p,q)$-forms and certain extensions $\mathscr{A}_X^{p,q}$ of them such that the corresponding Dolbeault complex…

The purpose of this paper is to point out a relation between the canonical sheaf and the intersection complex of a singular algebraic variety. We focus on the hypersurface case. Let $M$ be a complex manifold, $X\subset M$ a singular…

In this paper, we develop $L^2$ theory for Riemannian and Hermitian foliations on manifolds with basic boundary. We establish a decomposition theorem, various vanishing theorems, a twisted duality theorem for basic cohomologies and an…

We establish $L^2$ extension theorems for $\bar \partial$-closed $(0,q)$-forms with values in a holomorphic line bundle with smooth Hermitian metric, from a smooth hypersurface on a Stein manifold. Our result extends (and gives a new,…

Developing ideas of \cite{Fei}, we introduce canonical cosimplicial cohomology of meromorphic functions for infinite-dimensional Lie algebra formal series with prescribed analytic behavior on domains of a complex manifold $M$. Graded…