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\title{Irregularity of some toroidal compactifications}
\alttitle{Irrégularité de certaines compactifications toroïdales}

\author{\firstname{Ilya} \lastname{Karzhemanov}}
\address{Moscow Institute of Physics and Technology, 9 Institutskiy per., Dolgoprudny, Moscow Region, 141701, Russia}
\email{karzhemanov.iv@mipt.ru}

\thanks{The work was carried out at the Center for Pure Mathematics (MIPT) and supported by the State assignment project FSMG--2023--0013}

\subjclass{14J29, 14F17, 51M10}

\keywords{\kwd{Toroidal compactification} \kwd{irregularity} \kwd{Kähler hyperbolicity}}
\altkeywords{\kwd{Compactification toroïdale} \kwd{irrégularité} \kwd{hyperbolicité de Kähler}}

\begin{abstract}
We compute the irregularity of certain $2$-dimensional toroidal compactifications.
\end{abstract}

\begin{altabstract}
Nous calculons l'irrégularité de certaines compactifications toroïdales en dimension $2$.
\end{altabstract}

\begin{document}
%\input{CR-pagedemetas}
%\end{document}
\maketitle

\section{Introduction}\label{section:f}

\subsection{}\label{subsection:p-1}

The main objects of study of the present note are \emph{toroidal compactifications} $S$ of a complex ball quotient $U \coloneqq 
\mathbb{B}\slash\Gamma$ (see e.g.\ \cite{hol} for foundations). Recall that $S$ is a minimal (i.e.\ without $(-1)$-curves) projective surface and $U \subset S$ is a Zariski open subset with the following auxiliary data:
\begin{itemize}
\item $\mathbb{B}\subset\com\p^2$ is the unit ball and $\Gamma \subset \operatorname{PSU}(2,1)$ is a discrete subgroup\footnote{For all surfaces $S$, considered further, $\Gamma$ will be an \emph{arithmetic lattice}. This implies in particular that the rank of the homology group $H_1(U,\cel)$ is finite (see also~\cite[Corollary~4.5]{kas-san} or~\cite[Theorem~1.3]{di-sto}).} acting freely and properly discontinuously on $\mathbb{B}$;
\item the boundary divisor $D \coloneqq S\setminus U$ is a disjoint union of smooth elliptic curves (note that their self-intersection numbers are all negative).
\end{itemize}

We are interested in computing the \emph{irregularity} $q(S) \coloneqq h^1(S,\mathcal{O}_S)$ of the surface $S$ ---~under certain assumptions on the latter (see Section~\ref{subsection:p-2} and the discussion after Theorem~\ref{theorem:main} below for motivations).

\subsection{}\label{subsection:p-2}

Consider an Abelian surface $A$ and $\Bl(A) \coloneqq [\text{the blowup of $A$ at a finite set of distinct points}]$. Let $h \colon \Bl(A) \map A$ be the blowup map of distinct points on $A$ with the $(-1)$-curves $C_i \subset \Bl(A)$. We will require these to satisfy the following two properties:
\begin{enumerate}
\item\label{eq-1}
there exists a \emph{finite} morphism $f \colon S \map \Bl(A)$ (hence $q(S) \ge 2$) such that $\restr{f}{U}$ is \'etale;
\item\label{eq-2}
all preimages in $\mathbb{B}$ of the curves $f^{-1}(C_i) \cap U$ under the quotient morphism $\mathbb{B} \map \mathbb{B}\slash\Gamma$ have
\emph{infinitely many} connected components.
\end{enumerate}
Let us now formulate our main result.

\begin{theo}\label{theorem:main}
Under the assumptions~\eqref{eq-1} and~\eqref{eq-2} we have $q(S) = 2$.
\end{theo}

Note that setting $S \coloneqq Y_1$ in Theorem~\ref{theorem:main} one recovers the result from~\cite[2.2]{zuo}. These surfaces~$Y_1$ constitute a special case of examples from~\cite{hirz-1} (cf.~\cite{holz-2}) and \emph{all} the latter satisfy our preceding assumptions.

It is also interesting to compare the result of Theorem~\ref{theorem:main} with the case of \emph{compact} ball quotients. Namely, in~\cite{ishida-irreg} several smooth surfaces of general type, satisfying $c_1^2 = 3c_2 = 225$ and having irregularity $>2$, have been constructed. All these surfaces turned out to be the minimal resolutions of cyclic coverings of $\p^2$ ramified in configurations of lines (cf.~\cite{hirz}).

As for the non-compact ball quotients, let us recall that in~\cite[Theorem~1]{sto} (cf.~\cite[Theorem~5]{sto}) the author constructs a sequence $\{D_j\}$ of arithmetic quotients of $\mathbb{B}$, having finite volumes $\Vol D_j$ for all $j$ (so that $D_j$ are K\"ahler hyperbolic by the discussion in Section~\ref{subsection:f-1} below), arbitrarily large number of cusps (in fact their number grows like $(\Vol D_j)^{5/8}$) and topological Betti numbers $b_1(D_j)\gg(\Vol D_j)^{3/8}$ for $j \to \infty$ (compare with~\cite{agol}). All these $D_j$ arise from the so-called \emph{Deligne--Mostow orbifolds} (see~\cite[Section~4]{sto} for the precise constructions).

\begin{rema}\label{remark:momot-etc}
Note that in~\cite[Theorem~1.1]{mom}, assuming only that the Kodaira dimension $\Koddim(S)$ is less than $2$, the equality $q(S) = 2$ is claimed for \emph{any} toroidal compactification $S$. We should indicate however that some of the intermediate technical assertions stated in~\cite{mom} are wrong. Moreover, as it was pointed out in~\cite{di-sto}, the main result of~\cite{mom} is actually false because there exist such $S$ satisfying all the stated assumptions, but having $q(S) = 1$. Yet, it would still be interesting to repair (if possible) the method of~\cite{mom} in order to investigate those toroidal $S$, for which $\Koddim(S) < 2$ and $q(S) > 2$ (cf.~Section~\ref{subsection:f-2} below).
\end{rema}

The proof of Theorem~\ref{theorem:main} is given in Sections~\ref{subsection:f-2} and~\ref{subsection:f-4} by bringing the assumption $q(S) > 2$ to contradiction. Our argument relies on two ingredients: the vanishing $\mathcal{H}^1(U) = 0$ of $L^2$-cohomology (see Section~\ref{subsection:f-3}) and an explicit embedding $\mathbb{B} \subset \com^q$ via the Albanese map (cf.~\eqref{b-alb-q}).

\section{Proof of Theorem~\ref{theorem:main}}\label{section:p}

\subsection{K\"ahler hyperbolicity}\label{subsection:f-1}

We retain the notation of Section~\ref{section:f}. Let $\omega$ be a K\"ahler form on $S$ with at most logarithmic poles along $D$ (e.g.\ the one that represents the class of $c_1\parens[\big]{\mathcal{O}_S(1)} + D$ for some projective embedding $S
\subset \p^N$ and $\mathcal{O}_S(1) \coloneqq 
\restr{\mathcal{O}_{\p^N}(1)}{S}$). Then we have the following.

\begin{lemm}\label{theorem:fin-vol}
$\Vol U \coloneqq \int_U
\omega\wedge\overline{\omega} < \infty$.
\end{lemm}

\begin{proof}
Locally near every (smooth) component of $D$ one may set $U$ to coincide with $\mathbb{B}$ and $\restr{\omega}{U}$ be induced by the Poincar\'e metric. The claim now follows via the standard partition of unity argument (cf.~\cite[Section~1]{mum}).
\end{proof}

From Lemma~\ref{theorem:fin-vol} we readily obtain that $U$ is
\emph{complete} (with respect to the metric corresponding to $\restr{\omega}{U}$) ---~for all compact balls in $U$ have uniformly bounded volumes and so any diverging sequence of points $u_i \in U$ can have a limit only on the boundary $\partial U = D$. In particular, replacing~$U$ by $\mathbb{B}$ near $D$ (cf.~the proof of Lemma~\ref{theorem:fin-vol}) and using the completeness of Poincar\'e metric, we obtain that the sequence of $u_i$ is not Cauchy. Hence $U$ is complete.

Following~\cite{gro-kah-hyp} we call $U$ \emph{K\"ahler hyperbolic} ---~for the lift $\tilde{\omega}$ of the K\"ahler form $\restr{\omega}{U}$ to $\mathbb{B}$ is
\emph{$d$(bounded)} (cf.~\cite[0.1.B, 0.2.A, 0.3.A(b)]{gro-kah-hyp}). What makes this property interesting to us is Theorem~\ref{theorem:lef-van} below. The latter will be used in estimating the irregularity $q(S)$ of the surfaces satisfying the assumptions from Theorem~\ref{theorem:main}.

\subsection{\texorpdfstring{$L^2$}{L\^2}-cohomology}\label{subsection:f-3}

Next we consider the spaces of \emph{$L^2$-cohomology} $\mathcal{H}^p(U)$ consisting of harmonic (w.r.t.\ $\restr{\omega}{U}$) holomorphic $L^2$-forms on $U$ of degree $p \ge 0$. One identifies these $\mathcal{H}^p(U)$ with the subspaces of similarly defined (w.r.t.\ $\tilde{\omega}$) $\Gamma$-invariant forms on $\mathbb{B}$.

Here is the Lefschetz-type result we will need.

\begin{theo}[{see~\cite[1.2.B, 2.5, 2.5.A]{gro-kah-hyp}}]\label{theorem:lef-van}
$\mathcal{H}^1(U) = 0$.
\end{theo}

In addition, if $L^2\Omega^p$ denotes the sheaf of holomorphic $L^2$-forms on $U$ of degree $p$, then the following Hodge decomposition takes place (see~\cite[1.1.C]{gro-kah-hyp}):
\begin{equation}\label{hodge}
L^2\Omega^p = \mathcal{H}^p(U) \oplus
\overline{\dif L^2\Omega^{p-1}}\oplus\overline{\delta L^2\Omega^{p+1}}.
\end{equation}
Here $\dif$ is the usual K\"ahler differential, $\delta \coloneqq \pm {*}{\dif}{*}$, $*$ is the Hodge star operator acting on $H^{\bullet}(U)$, and the last two direct summands in~\eqref{hodge} are the $L^2$-closures of respective images. A similar decomposition (with~\eqref{hodge} being its $\Gamma$-invariant part) also holds for the $L^2$-forms on $\mathbb{B}$.

\subsection{Universal cover of \texorpdfstring{$U$}{U} when \texorpdfstring{$q(S) > 2$}{q(S) > 2}}\label{subsection:f-2}

From this moment on we will assume that $q \coloneqq q(S) > 2$ (recall that initially $q(S) \ge 2$ by the discussion in Section~\ref{subsection:p-2}).

Let $\alb \colon S \map \Alb(S)$ be the Albanese morphism. Consider the preimage $\Delta$ of $\alb(D)$ in the universal cover $\com^q$ of $\Alb(S)$.

\begin{lemm}\label{theorem:u-q-sim-con}
$\pi_1(\com^q\setminus\Delta) =
\{1\}$.
\end{lemm}

\begin{proof}
Note that $\Delta$ is a countable union $\bigcup_i L_i$ of \emph{affine lines} $L_i \subset \com^q$. Indeed, this follows from the fact that $D$ is a disjoint union of elliptic curves, as we have assumed in Section~\ref{subsection:p-1}.

Further, write $\com^q\setminus\Delta = \bigcap_k \parens[\big]{\com^q \setminus{\bigcup_{i = 1}^k L_i}}$, which implies (by definition of the fundamental group and universal cover) that
\begin{equation}\label{pi-s-c-q}
\pi_1(\com^q\setminus\Delta) = \varprojlim_k G_k.
\end{equation}
Here the groups $G_k \coloneqq \pi_1\parens[\big]{\com^q \setminus{\bigcup_{i = 1}^k L_i}}$ form a projective system with respect to the natural surjections $G_{k+1}
\twoheadrightarrow G_k$ for all $k\ge 1$ (cf.~\cite[Proposition~2.10,~(2.10.1)]{kol-shaf-m}).

Finally, recall that according to~\cite[Theorem~2.14,~(2.14.1)]{kol-shaf-m} if $X$ is a normal quasi-projective variety with a pencil $\lvert H \rvert$ of hypersurfaces and a base point $x \in X$, whose general member $H_t \ni x$ is connected, then the natural homomorphism $\pi_1(H_t) \to \pi_1(X)$ is surjective. Let us apply this result to show that $G_k = \{1\}$ for all $k$. Namely, put $X \coloneqq \com^q \setminus{\bigcup_{i = 1}^k L_i}$ and $H_t \coloneqq X \cap
\com^{q-1}$, where $\com^{q-1} \subset \com^q$ is a general affine hyperplane. We have $H_t = \com^{q-1} \setminus [\text{a finite set of points $L_i \cap \com^{q-1}$}]$, which implies that $\pi_1(H_t) = \pi_1(\com^{q-1})$, since $q > 2$ by assumption. Thus we get that $\pi_1(H_t) = \{1\}$ surjects onto $\pi_1(X) = G_k$. Then $\pi_1(\com^q\setminus\Delta) = \{1\}$ as well according to~\eqref{pi-s-c-q}.
\end{proof}

Identify the origin $0 \in \mathbb{B}$ with its image in $U$ under the universal covering $\mathbb{B} \map U = \mathbb{B} \slash
\Gamma$.

Let $\gamma \in H_1(U,\com)$ be any element, which we treat as a loop on $U$ passing through $0$, and let $\tilde{\gamma}$ be some lift of $\gamma$ to $\mathbb{B}$. Let also $\omega_1,\dots,\omega_q$ be $\Gamma$-invariant closed holomorphic $1$-forms on $\mathbb{B}$ that give a basis for $H^1(S,\com)$ when restricted to $U \subset S$. Notice that by construction all $\omega_i$ are \emph{bounded} $1$-forms on $\mathbb{B}$.

\begin{figure}[h]
\includegraphics{pic}
\caption{$2q$ $1$-cycles on $U$.}\label{fig-1}
\end{figure}

Choose a splitting for the natural surjection $H_1(U,\cel)
\twoheadrightarrow H_1(S,\cel) = \cel^{2q} \oplus \mathrm{torsion}$ (cf.~Figure~\ref{fig-1}). Then from Lemma~\ref{theorem:u-q-sim-con} and definition of $\alb$ we obtain the following
\emph{analytic} morphism:
\begin{equation}\label{b-alb-q}
x \in \mathbb{B} \mapsto \parens*{\int_0^x
\omega_1,\dots,\int_0^x\omega_q}\in \com^q_{\Delta} \coloneqq 
\com^q\setminus\Delta
\end{equation}
(here the ring of holomorphic functions on $\com^q_{\Delta}
\subset \com^q$ is generated by the affine coordinates coming from $\com^q$ because $\pi_1(\com^q_{\Delta}) = \{1\}$). More precisely, the integrals $\int_0^x \omega_i$ along a path $[0,x]$ joining $0$ and a given point $x$ are defined modulo various $\int_{\tilde{\gamma}}\omega_i$, with $\gamma\in H_1(S,\cel) \subseteq H_1(U,\cel)$. But the latter integrals are all zero for $\pi_1(\mathbb{B}) = \{1\}$ and $\omega_i$ being bounded along the $[0,x]$.

We observe next that the morphism
\[
\restr{\alb}{U} \colon U \map
\com^q_{\Delta}\slash H_1(S,\cel) =
\Alb(S)\setminus\alb(D)
\]
fits into a commutative diagram of analytic spaces and morphisms
\begin{equation}\label{b-alb-q-123}
\xymatrix{
\mathbb{B}\ar@{->}[d]\ar@{->}[r]^{\alb_{\mathbb{B}}}&\com^q_{\Delta}\ar@{->}[d]\\
U\ar@{->}[r]&\Alb(S)\setminus\alb(D).}
\end{equation}
Here the vertical arrows are the universal coverings of $U$ and $\Alb(S)\setminus\alb(D)$, whereas the horizontal ones are~\eqref{b-alb-q} and $\restr{\alb}{U}$, respectively.

\begin{rema}\label{remark:com-on-con-asghhj}
Note that the morphism $U \map \Alb(S)\setminus\alb(D)$ in~\eqref{b-alb-q-123} depends on the splitting of $H_1(U,\cel) \twoheadrightarrow H_1(S,\cel)$ and need not extend to $S$ (i.e.\ coincide with $\restr{\alb}{U}$) if the codimension of $\alb(D)$ is $1$ (and hence $q = 2$). Indeed, we have $\pi_1(\com^q_{\Delta}) \ne \{1\}$ in this case, so that~\eqref{b-alb-q} is not analytic ---~the ring of holomorphic functions on $\com^2_{\Delta}$ is \emph{larger} than that on $\com^2$.
\end{rema}

\begin{prop}\label{theorem:alb-u-fin-et}
The induced morphism $\restr{\alb}{U} \colon U \map \alb(U)$ is finite and smooth.
\end{prop}

\begin{proof}
Note that the morphism $h \circ f$ factors through $\alb$ by functoriality (cf.~Section~\ref{subsection:p-2}). Hence it suffices to show that none of the curves $f^{-1}(C_i) \subset S$ is contracted by $\alb$. Suppose the contrary. Then, restricting $\alb_{\mathbb{B}}$ to the preimages of $f^{-1}(C_i) \cap U$ in $\mathbb{B}$, we find a set $\mathfrak{S} \coloneqq [\text{disjoint union of infinitely}$ $\text{many curves in $\mathbb{B}$}]$ such that all the (holomorphic) components $f_i$ of $\alb_{\mathbb{B}}$ are constant on $\mathfrak{S}$, $1 \le i \le q$.

On the other hand, it follows from the construction (see~\eqref{b-alb-q}) that $\alb_{\mathbb{B}}$ extends to the closure $\overline{\mathbb{B}} \subset \p^2$, with $\mathfrak{S}$ having accumulation points on the boundary $\overline{\mathbb{B}}\setminus\mathbb{B}$. In particular, we find that all the $f_i$ must be identically constant, which is impossible because $\dim\alb_{\mathbb{B}}(\mathbb{B}) =
\dim \alb(U) = 2$.

The proof of Proposition~\ref{theorem:alb-u-fin-et} is complete.
\end{proof}

\begin{coro}\label{theorem:im-is-b-quot}
$\restr{\alb}{U}$ is an isomorphism (onto its image).
\end{coro}

\begin{proof}
This follows from Proposition~\ref{theorem:alb-u-fin-et} and~\eqref{b-alb-q-123}. Indeed, the morphism $\alb_{\mathbb{B}}$ must also be finite and smooth, which implies that $\alb_{\mathbb{B}}(\mathbb{B}) \subset
\com^q$ is a \emph{contractible subvariety} (cf.~\eqref{b-alb-q}). Hence $\alb_{\mathbb{B}}$ (and $\restr{\alb}{U}$ as well) is of topological degree~$1$.
\end{proof}

\subsection{Final step: the use of vanishing \texorpdfstring{$\mathcal{H}^1(U) = 0$}{H\^1(U) = 0}}\label{subsection:f-4}

It follows from (the proof of) Corollary~\ref{theorem:im-is-b-quot} that $\alb_{\mathbb{B}}$ is an isomorphism onto its image. Thus we may identify $\mathbb{B}$ (resp.\ $U$) with an analytic subset in $\com^q$ (resp.\ with $\alb(U) \subset\Alb(S)$). Note also that $\mathbb{B} = \alb_{\mathbb{B}}(\mathbb{B}) \subset
\com^q$ is a \emph{bounded} subset by construction of $\alb_{\mathbb{B}}$. In particular, for appropriate \emph{affine} coordinates $x_1,\dots,x_q$ on $\com^q
\supset \mathbb{B}$ one may put $x_1,x_2$ to be analytic coordinates on $\mathbb{B}$, so that $x_i = x_i(x_1,x_2)$ are some holomorphic functions in $x_1,x_2$ for all $i\ge 3$.

Further, since $\pi_1(U) = \Gamma$ (cf.~Section~\ref{subsection:p-1}), it follows from~\eqref{b-alb-q-123} that the group $\Gamma
\subseteq H_1(S,\cel)$ acts on~$\mathbb{B}$ via the $\cel$-shifts,
\begin{equation}\label{gamma-act-shif}
\gamma \colon x_i \mapsto x_i + c_i(\gamma), \quad \gamma\in\Gamma,
\end{equation}
with certain $c_i(\gamma)\in\cel$ and all $i$. We can also assume $c_1(\gamma)\ne 0$ for some $\gamma\in\Gamma$ because $x_1, x_2$ are coordinates on $\mathbb{B}$, where $\Gamma$ acts
\emph{non-trivially}. Let us bring this to contradiction (compare with~\cite{law-yau}).

Consider the $1$-form $v \coloneqq \dif x_1$. It is $\Gamma$-invariant by~\eqref{gamma-act-shif} and its descent to $U$ is $L^2$ w.r.t.\ the metric corresponding to $\restr{\omega}{U}$ (the $L^2$-norm of $v$ equals $\Vol U$ by definition). Furthermore, since $\dif v = 0$ and $\mathcal{H}^1(U) = 0$ by Theorem~\ref{theorem:lef-van}, from~\eqref{hodge} we obtain that $v = \lim_{i \to
\infty} \dif f_i$ for some $\Gamma$-invariant holomorphic $L^2$-functions $f_i$ on $\mathbb{B}$. In particular, since $v = \dif x_1$, we obtain that $x_1 = \lim_{i \to \infty} f_i$ is $\Gamma$-invariant, which contradicts $c_1(\gamma)\ne 0$.

The proof of Theorem~\ref{theorem:main} is finished.

\section*{Acknowledgments}

I am grateful to anonymous referee for valuable suggestions.

\section*{Declaration of interests}

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

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