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Mats Andersson

Publications and source records attributed to Mats Andersson.

At least 19 recordsLinked to original sources
arXiv2025

Segre forms of singular metrics on vector bundles and Lelong numbers

Let $E\to X$ be a holomorphic vector bundle. We consider a class of a singular Hermitian metrics on $E$ with analytic singularities that contains all Griffiths negative such metrics. One can define, given a smooth reference metric $h_0$, a current $s(E,h,h_0)$ called the associated Segre form, which defines the expected Bott-Chern class and coincides with the usual Segre form of $h$ where it is smooth. We prove that $s(E,h,h_0)$ is the limit of the Segre forms of a sequence of smooth metrics if the metric is smooth outside the degeneracy locus, and in general as a limit of Segre forms of metrics with empty degeneracy locus. One can also define an associated Chern form $c(E,h,h_0)$. We prove that the Lelong numbers of $s(E,h,h_0)$ and $c(E,h,h_0)$ are integers if the singularities are integral, and non-negative for $s(E,h,h_0)$.

math.CV↗
arXiv2024

Poincaré-Lelong type formulas and Segre numbers

Let $E$and $F$ be Hermitian vector bundles over a complex manifold $X$ and let $g\colon E\to F$ be a holomorphic morphism. We prove a Poincaré-Lelong type formula with a residue term $M^g$. The currents $M^g$ so obtained have an expected functorial property. We discuss various applications: If $F$ has a trivial holomorphic subbundle of rank $r$ outside the analytic set $Z$, then we get currents with support on $Z$ that represent the Bott-Chern classes $\hat c_k(E)$ for $k >\rank E-r$. We also consider Segre and Chern forms associated with certain singular metrics on $F$. The multiplicities (Lelong numbers) of the various components of $M^g$ only depend on the cokernel of the adjoint sheaf morphism $g^*$. This leads to a notion of distinguished varieties and Segre numbers of an arbitrary coherent sheaf, generalizing these notions, in particular the Hilbert-Samuel multiplicity, in case of an ideal sheaf.

math.CV↗
arXiv2023

Norm estimates for the $\bar\partial$-equation on a non-reduced space

We study norm-estimates for the $\bar\partial$-equation on non-reduced analytic spaces. Our main result is that on a non-reduced analytic space, which is Cohen-Macaulay and whose underlying reduced space is smooth, the $\bar\partial$-equation for $(0,1)$-forms can be solved with $L^p$-estimates.

math.CV↗
arXiv2022

Non-pluripolar energy and the complex Monge-Ampère operator

Given a domain $Ω\subset \mathbf C^n$ we introduce a class of plurisubharmonic (psh) functions $\mathcal G(Ω)$ and Monge-Ampère operators $u\mapsto [dd^c u]^p$, $p\leq n$, on $\mathcal G(Ω)$ that extend the Bedford-Taylor-Demailly Monge-Ampère operators. Here $[dd^c u]^p$ is a closed positive current of bidegree $(p,p)$ that dominates the non-pluripolar Monge-Ampère current $\langle dd^c u\rangle^p$. We prove that $[dd^c u]^p$ is the limit of Monge-Ampère currents of certain natural regularizations of $u$. On a compact Kähler manifold $(X, ω)$ we introduce a notion of non-pluripolar energy and a corresponding finite energy class $\mathcal G(X, ω)\subset \text{PSH}(X, ω)$ that is a global version of $\mathcal G(Ω)$. From the local construction we get global Monge-Ampère currents $[dd^c φ+ ω]^p$ for $φ\in \mathcal G(X,ω)$ that only depend on the current $dd^c φ+ ω$. The limits of Monge-Ampère currents of certain natural regularizations of $φ$ can be expressed in terms of $[dd^c φ+ ω]^j$ for $j\leq p$. We get a mass formula involving the currents $[dd^c φ+ω]^p$ that describes the loss of mass of the non-pluripolar Monge-Ampère measure $\langle dd^c φ+ω\rangle^n$. The class $\mathcal G(X, ω)$ includes $ω$-psh functions with analytic singularities and the class $\mathcal E(X, ω)$ of $ω$-psh functions of finite energy and certain other convex energy classes, although it is not convex itself.

math.CV↗
arXiv2021

On proper intersections on a singular analytic space

Given a reduced analytic space $Y$ we introduce a class of {\it nice} cycles, including all effective $\mathbb{Q}$-Cartier divisors. Equidimensional nice cycles that intersect properly allow for a natural intersection product. Using $\bar{\partial}$-potentials and residue calculus we provide an intrinsic way of defining this product. The intrinsic definition makes it possible to prove global formulas. In case $Y$ is smooth all cycles are differences of nice cycles, and so we get a new way to define classical proper intersections.

math.CV↗
arXiv2021

On non-proper intersections and local intersection numbers

Given pure-dimensional (generalized) cycles $μ_1$ and $μ_2$ on a complex manifold $Y$ we introduce a product $μ_1\diamond_{Y} μ_2$ that is a generalized cycle whose multiplicities at each point are the local intersection numbers at the point. % If $Y$ is projective, then given a very ample line bundle $L\to Y$ we define a product $μ_1\bl μ_2$ whose multiplicities at each point also coincide with the local intersection numbers. In addition, provided that $μ_1$ and $μ_2$ are effective, this product satisfies a Bézout inequality. If $i\colon Y\to \Pk^N$ is an embedding such that $i^*\Ok(1)=L$, then $μ_1\bl μ_2$ can be expressed as a mean value of Stückrad-Vogel cycles on $\Pk^N$. There are quite explicit relations between $\di_Y$ and $\bl$.

math.CV↗
arXiv2020

A 6G White Paper on Connectivity for Remote Areas

In many places all over the world rural and remote areas lack proper connectivity that has led to increasing digital divide. These areas might have low population density, low incomes, etc., making them less attractive places to invest and operate connectivity networks. 6G could be the first mobile radio generation truly aiming to close the digital divide. However, in order to do so, special requirements and challenges have to be considered since the beginning of the design process. The aim of this white paper is to discuss requirements and challenges and point out related, identified research topics that have to be solved in 6G. This white paper first provides a generic discussion, shows some facts and discusses targets set in international bodies related to rural and remote connectivity and digital divide. Then the paper digs into technical details, i.e., into a solutions space. Each technical section ends with a discussion and then highlights identified 6G challenges and research ideas as a list.

eess.SP↗
arXiv2020

A pointwise norm on a non-reduced analytic space

Let $X$ be a possibly non-reduced space of pure dimension. We introduce an essentially intrinsic pointwise Hermitian norm on smooth $(0,*)$-forms, in particular on holomorphic functions, on $X$. We prove that the space of holomorphic functions is complete with respect to the natural topology induced by this norm.

math.CV↗
arXiv2020

Global representation of Segre numbers by Monge-Ampère products

On a reduced analytic space $X$ we introduce the concept of a generalized cycle, which extends the notion of a formal sum of analytic subspaces to include also a form part. We then consider a suitable equivalence relation and corresponding quotient $\mathcal{B}(X)$ that we think of as an analogue of the Chow group and a refinement of de Rham cohomology. This group allows us to study both global and local intersection theoretic properties. We provide many $\mathcal{B}$-analogues of classical intersection theoretic constructions: For an analytic subspace $V\subset X$ we define a $\mathcal{B}$-Segre class, which is an element of $\mathcal{B}(X)$ with support in $V$. It satisfies a global King formula and, in particular, its multiplicities at each point coincide with the Segre numbers of $V$. When $V$ is cut out by a section of a vector bundle we interpret this class as a Monge-Ampère-type product. For regular embeddings we construct a $\mathcal{B}$-analogue of the Gysin morphism.

math.CV↗
arXiv2020

The $\bar\partial$-equation for $(p,q)$-forms on a non-reduced analytic space

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 is exact, i.e., the $\bar\partial$-equation is locally solvable in $\mathscr{A}_X$. The sheaves $\mathscr{A}_X^{p,q}$ are modules over the smooth forms, in particular, they are fine sheaves. We also introduce certain sheaves $\mathscr{B}_X^{n-p,n-q}$ of currents on $X$ that are dual to $\mathscr{A}_X^{p,q}$ in the sense of Serre duality. More precisely, we show that the compactly supported Dolbeault cohomology of $\mathscr{B}^{n-p,n-q}(X)$ in a natural way is the dual of the Dolbeault cohomology of $\mathscr{A}^{p,q}(X)$.

math.CV↗
arXiv2019

Nonproper intersection products and generalized cycles

In this article we develop intersection theory in terms of the $\mathcal{B}$-group of a reduced analytic space. This group was introduced in a previous work as an analogue of the Chow group; it is generated by currents that are direct images of Chern forms and it contains all usual cycles. However, contrary to Chow classes, the $\mathcal{B}$-classes have well-defined multiplicities at each point. We focus on a $\mathcal{B}$-analogue of the intersection theory based on the Stückrad-Vogel procedure and the join construction in projective space. Our approach provides global $\mathcal{B}$-classes which satisfy a Bézout theorem and have the expected local intersection numbers. An essential feature is that we take averages, over various auxiliary choices, by integration. We also introduce $\mathcal{B}$-analogues of more classical constructions of intersections using the Gysin map of the diagonal. These constructions are connected via a $\mathcal{B}$-variant of van Gastel's formulas. Furthermore, we prove that our intersections coincide with the classical ones on cohomology level.

math.AG↗
arXiv2019

The flatness of the $\Ok$-module of smooth functions and integral representation

We give a proof of the well-known fact that the $\Ok$-module $\E$ of smooth functions is flat by means of residue theory and integral formulas. A variant of the proof gives a related statement for classes of functions of lower regularity. We also prove a Briançon-Skoda type theorem for ideals of the form $\E a$, where $a$ is an ideal in $\Ok$.

math.CV↗
arXiv2019

Segmenting Potentially Cancerous Areas in Prostate Biopsies using Semi-Automatically Annotated Data

Gleason grading specified in ISUP 2014 is the clinical standard in staging prostate cancer and the most important part of the treatment decision. However, the grading is subjective and suffers from high intra and inter-user variability. To improve the consistency and objectivity in the grading, we introduced glandular tissue WithOut Basal cells (WOB) as the ground truth. The presence of basal cells is the most accepted biomarker for benign glandular tissue and the absence of basal cells is a strong indicator of acinar prostatic adenocarcinoma, the most common form of prostate cancer. Glandular tissue can objectively be assessed as WOB or not WOB by using specific immunostaining for glandular tissue (Cytokeratin 8/18) and for basal cells (Cytokeratin 5/6 + p63). Even more, WOB allowed us to develop a semi-automated data generation pipeline to speed up the tremendously time consuming and expensive process of annotating whole slide images by pathologists. We generated 295 prostatectomy images exhaustively annotated with WOB. Then we used our Deep Learning Framework, which achieved the $2^{nd}$ best reported score in Camelyon17 Challenge, to train networks for segmenting WOB in needle biopsies. Evaluation of the model on 63 needle biopsies showed promising results which were improved further by finetuning the model on 118 biopsies annotated with WOB, achieving F1-score of 0.80 and Precision-Recall AUC of 0.89 at the pixel-level. Then we compared the performance of the model against 17 biopsies annotated independently by 3 pathologists using only H\&E staining. The comparison demonstrated that the model performed on a par with the pathologists. Finally, the model detected and accurately outlined existing WOB areas in two biopsies incorrectly annotated as totally WOB-free biopsies by three pathologists and in one biopsy by two pathologists.

cs.CV↗
arXiv2018

Multi-Resolution Networks for Semantic Segmentation in Whole Slide Images

Digital pathology provides an excellent opportunity for applying fully convolutional networks (FCNs) to tasks, such as semantic segmentation of whole slide images (WSIs). However, standard FCNs face challenges with respect to multi-resolution, inherited from the pyramid arrangement of WSIs. As a result, networks specifically designed to learn and aggregate information at different levels are desired. In this paper, we propose two novel multi-resolution networks based on the popular `U-Net' architecture, which are evaluated on a benchmark dataset for binary semantic segmentation in WSIs. The proposed methods outperform the U-Net, demonstrating superior learning and generalization capabilities.

cs.CV↗
arXiv2018

Global Koppelman formulas on (singular) projective varieties

Let $i\colon X\to \Pk^N$ be a projective manifold of dimension $n$ embedded in projective space $\Pk^N$, and let $L$ be the pull-back to $X$ of the line bundle $\Ok_{\Pk^N}(1)$. We construct global explicit Koppelman formulas on $X$ for smooth $(0,*)$-forms with values in $L^s$ for any $s$. %The formulas are intrinsic on $X$. The same construction works for singular, even non-reduced, $X$ of pure dimension, if the sheaves of smooth forms are replaced by suitable sheaves $\A_X^*$ of $(0,*)$-currents with mild singularities at $X_{sing}$. In particular, if $s\ge \reg X -1$, where $\reg X$ is the Castelnuovo-Mumford regularity, we get an explicit %%% representation of the well-known vanishing of $H^{0,q}(X, L^{s-q})$, $q\ge 1$. Also some other applications are indicated.

math.CV↗
arXiv2018

The $\bar{\partial}$-equation on a non-reduced analytic space

Let $X$ be a, possibly non-reduced, analytic space of pure dimension. We introduce a notion of $\overline{\partial}$-equation on $X$ and prove a Dolbeault-Grothendieck lemma. We obtain fine sheaves $\mathcal{A}_X^q$ of $(0,q)$-currents, so that the associated Dolbeault complex yields a resolution of the structure sheaf $\mathscr{O}_X$. Our construction is based on intrinsic semi-global Koppelman formulas on $X$.

math.CV↗