%~Mouliné par MaN_auto v.0.34.0 (b0f8651d) 2025-07-04 14:55:20
\documentclass[CRMATH,Unicode,XML,biblatex,published]{cedram}

\TopicFR{Géométrie algébrique, Théorie des singularités}
\TopicEN{Algebraic geometry, Singularity theory}

\addbibresource{CRMATH_Sabbah_20250465.bib}

\DeclareMathOperator{\ICC}{IC}

%%%---------------------------------------------------------------------------

%%% ARROWS

%%% \to
\renewcommand*{\to}{\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}}

%%%---------------------------------------------------------------------------

%%% DELIMITERS

\usepackage{mathtools}
\DeclarePairedDelimiter{\parens}{\lparen}{\rparen}
\DeclarePairedDelimiter{\braces}{\{}{\}}
%\DeclarePairedDelimiter{\abs}{\lvert}{\rvert}
%\DeclarePairedDelimiter{\norm}{\lVert}{\rVert}
%\DeclarePairedDelimiter{\tnorm}{\VERT}{\VERT}
%\DeclarePairedDelimiter{\bracks}{[}{]}
%\DeclarePairedDelimiter{\bbracks}{\llbracket}{\rrbracket}
%\DeclarePairedDelimiter{\angles}{\langle}{\rangle}
%\DeclarePairedDelimiter{\aangles}{\langle\!\langle}{\rangle\!\rangle}
%\DeclarePairedDelimiter{\pparens}{(\!(}{)\!)}

%%%---------------------------------------------------------------------------

\newcommand{\rH}{{\scriptscriptstyle\mathrm{H}}}
\newcommand{\QQ}{\mathbb Q}
\newcommand{\cK}{\mathcal K}
\newcommand{\cQ}{\mathcal Q}
\newcommand{\QH}{\QQ^\rH}

\graphicspath{{./figures/}}

\newcommand*{\mk}{\mkern -1mu}
\newcommand*{\Mk}{\mkern -2mu}
\newcommand*{\mK}{\mkern 1mu}
\newcommand*{\MK}{\mkern 2mu}

\hypersetup{urlcolor=purple, linkcolor=blue, citecolor=red}

\newcommand*{\relabel}{\renewcommand{\labelenumi}{(\theenumi)}}
\newcommand*{\romanenumi}{\renewcommand*{\theenumi}{\roman{enumi}}\relabel}
\newcommand*{\Romanenumi}{\renewcommand*{\theenumi}{\Roman{enumi}}\relabel}
\newcommand*{\alphenumi}{\renewcommand*{\theenumi}{\alph{enumi}}\relabel}
\newcommand*{\Alphenumi}{\renewcommand*{\theenumi}{\Alph{enumi}}\relabel}

%%%---------------------------------------------------------------------------

\makeatletter
\def\@setafterauthor{%
  \vglue3mm%
\begingroup\hsize=12cm\advance\hsize\abstractmarginL\raggedright
\noindent
   \leftskip\abstractmarginL
  \normalfont\Small
  \@afterauthor\par
\endgroup
\vskip2pt plus 3pt minus 1pt
}
\makeatother

%%%---------------------------------------------------------------------------

\title{Corrigendum to ``Hodge--Lyubeznik numbers''}
\alttitle{Corrigendum à \og Nombres de Hodge--Lyubeznik \fg}

\Relation[corrige]{10.5802/crmath.724}
\CDRsetmeta{articletype}{erratum}

\author{\firstname{Ricardo} \lastname{García López}}
\address{Departament de Matemàtiques i Informàtica, Universitat de Barcelona, Gran Vía 585, 08007~Barcelona, Spain}
\email{ricardogarcia@ub.edu}

\author{\firstname{Claude} \lastname{Sabbah}}
\address{CMLS, CNRS, École polytechnique, Institut Polytechnique de Paris, 91128 Palaiseau cedex, France}
\email{claude.sabbah@polytechnique.edu}

\thanks{The first author received partial support from grant PID2022-137283NB-C22 funded by MCIN/AEI/10.13039/501100011033}
\CDRGrant[MCIN/AEI/10.13039/501100011033]{PID2022-137283NB-C22}

\subjclass{13D45, 14B15, 32S35}

\keywords{\kwd{Hodge numbers} \kwd{Lyubeznik numbers} \kwd{mixed Hodge modules}}
\altkeywords{\kwd{Nombres de Hodge} \kwd{nombres de Lyubeznik} \kwd{modules de Hodge mixtes}}

\begin{abstract}
A Tate twist is missing in the exact sequence of Example~(1), leading to incorrect formulas for some Hodge--Lyubeznik numbers, as noticed by Bradley Dirks. We provide the correct formulas.
\end{abstract}

\begin{altabstract}
Un twist de Tate est manquant dans la suite exacte de l'Exemple~(1), induisant des formules incorrectes pour certains nombres de Hodge--Lyubeznik, comme remarqué par Bradley Dirks. Nous donnons les formules correctes.
\end{altabstract}

\dateposted{2025-07-15}
\begin{document}

\maketitle

The exact sequence of~\cite[Example~(1)]{G-S25} should read
\[
H^{s-3}(M) \overset{L}{\longrightarrow} H^{s-1}(M)(1) \to H^s_{\{0\}}(X)(1) \to H^{s-2}(M) \overset{L}{\longrightarrow} H^s(M)(1).
\]
The Tate twist in the middle term was mistakenly omitted, leading to incorrect formulas for the Hodge--Lyubeznik numbers computed by means of it. The consequences of this error were noticed by Bradley Dirks. With the exact same reasoning, the formulas for the Hodge--Lyubeznik numbers are:
\begin{itemize}
\item In Example~(1):
\bgroup
\begin{enumerate}
\item\label{enum:ci}
$\lambda^{p,q}_{0,s}(\mathcal{O}_{X,0})=h^{-p,-q}\parens[\big]{H^{s-1}(M)_\mathrm{prim}}$ for $2\leq s \leq d-1$;
\item\label{enum:cii}
$\lambda^{p,q}_{r,d}(\mathcal{O}_{X,0})= h^{p+d-1, q+d-1}\parens[\big]{H^{d+r-2}(M)_\mathrm{coprim}}$ for $2 \leq r \leq d$;
\item\label{enum:ciii}
all other $\lambda^{p,q}_{r,s}(\mathcal{O}_{X,0})$ vanish.
\end{enumerate}
\egroup
\item In Example~(2):
\[
\lambda_{0,4}^{-p,-q}(A_i)=h^{p,q}\parens[\big]{H^3(M_i)_\mathrm{prim}}=h^{p,q}\parens[\big]{H^3(M_i)}-h^{p-1,q-1}\parens[\big]{H^1(M_i)}.
\]
In particular, if $N_1=X_3 (70, 16, 16, 14, 7, 6)$ we obtain, according to~\cite{WYW},
\[
\lambda^{-1,-2}_{0,4}(A_1)= h^{1,2}\parens[\big]{H^3(N_1)}=3365330286081,
\]
while if $N_2=X_3 (56, 49, 8, 6, 5, 4, 4)$,
\[
\lambda^{-1,-2}_{0,4}(A_2)= h^{1,2}\parens[\big]{H^3(N_2)}=3343868254721.
\]
\item In Remark~(1) on page~220,
\[
h^{p,q}\cK(-n) = h^{p,q}\parens[\big]{H_{\{0\}}^d(X)}=h^{p,q}\parens[\big]{H^{d-1}(M)_\mathrm{prim}},
\]
and $\cK(-n)$ is pure of weight $d-1$. It follows that $\cQ(n)$ is pure of weight $1-d$ and $h^{p,q}\cQ(n)=h^{-p,-q}\parens[\big]{H^{d-1}(M)_\mathrm{prim}}$.
\end{itemize}

We take this opportunity to correct a typo: on page~218, two lines after~(6), one should read ``its restriction to $U$ is $\QH_U[d](n)$''.

\section*{Acknowledgments}

We thank Bradley Dirks for noticing erroneous formulas and the typo.

\section*{Declaration of interests}

The authors do not work for, advise, own shares in, or receive funds from any organization that could benefit from this article, and have declared no affiliations other than their research organizations.

\printbibliography

\end{document}