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Dieudonné, Université Côte d’Azur, CNRS UMR 7351,\\ Parc Valrose 06108 Nice Cedex 2, France} \email{gachet@unice.fr} \urladdr{https://cecilegachet.github.io/} \begin{abstract} Let $A$ be an abelian variety, and $G\subset\mathrm{Aut}(A)$ a finite group acting freely in codimension two. We~discuss whether the singular quotient $A/G$ admits a resolution that is a Calabi-Yau manifold. While Oguiso constructed two examples in dimension 3, we~show that there are none in dimension 4. We~also classify up to isogeny the possible abelian varieties $A$ in arbitrary dimension. \end{abstract} \subjclass{14J32, 14K22, 14E15, 14L30} \keywords{Crepant resolution, abelian variety with complex multiplication, Calabi-Yau manifold} \altkeywords{Résolution crépante, variété abélienne à multiplication complexe, variété de Calabi-Yau} \alttitle{Quotients finis de variétés abéliennes admettant une résolution de Calabi-Yau} \begin{altabstract} Soit $A$ une variété abélienne et $G$ un groupe fini agissant librement en codimension $2$ par automorphismes sur $A$. On s’intéresse ici aux conditions d'existence d’une résolution du quotient singulier $A/G$ qui soit une variété de Calabi-Yau. Tandis qu'en dimension $3$, deux exemples de quotients admettant une telle résolution ont été construits par Oguiso dans un article de 1994, on montre ici qu’aucun quotient de la sorte n’existe en dimension $4$. En dimension quelconque, on classifie les variétés abéliennes $A$ susceptibles d’admettre des quotients de la sorte, à isogénie près. \end{altabstract} \maketitle \tableofcontents \mainmatter \section{Introduction} Since singularities are a byproduct of the Minimal Model Program, studying singular varieties with trivial canonical class, or singular $K$-trivial varieties, is an important question in the birational classification of complex algebraic varieties. From this point of view, the recent generalization of the Beauville-Bogomolov decomposition theorem for smooth $K$-trivial varieties (\cite{Beauv84}) to klt $K$-trivial varieties (\cite{GGK,Druel,HoeringPeternell,BGL}) is highly relevant. It indeed establishes that, after a finite quasiétale cover, any klt $K$-trivial variety is a product of a smooth abelian variety, some irreducible holomorphic symplectic varieties with canonical singularities, also called hyperkähler varieties, and some Calabi-Yau varieties with canonical singularities. These three main families of $K$-trivial varieties are the subject of large, mostly disjoint realms of the literature, ranging from the well-known theory of abelian varieties (exposed notably in the reference books \cite{BirkLang,Shimura}), through the thriving study of hyperkähler varieties (see \cite{DebarreHK,AmVerbSurvey,Kamenova} for surveys), to the unruly ``Zoo of Calabi-Yau varieties'', populated by a huge amount of examples (\cite{KreuzerSkarke1998,KreuzerSkarke2000} for K3 surfaces and Calabi-Yau threefolds embedded as hypersurfaces in toric varieties only), and whose boundedness is yet not established (see \cite{WilsonPic2, WilsonPic3, ChenDiCerboHanSvaldi,DiCerboSvaldi,BirkarDiCerboSvaldi} for recent breakthroughs). A new feature appearing in the context of singular $K$-trivial varieties is that birational $K$-trivial varieties may have different Beauville-Bogomolov decomposition types. For example, Kummer surfaces are K3 surfaces, but also minimal resolutions of finite quasiétale quotients of abelian surfaces. Similar examples of dimension $3$ exist, as in \cite{III0}, as well as higher dimensional examples, \cf \cite{CynkHulek,CynkSchuett,PR,AndreaWis,Burek}. In arbitrary dimension, it~is known that a crepant resolution or terminalization only changes the type of a klt $K$-trivial variety if its decomposition entails an abelian factor (\cite[Prop.\,4.10]{Druel}).\footnote{It reflects the more general fact that the Beauville-Bogomolov decomposition type of a klt $K$\nobreakdash-trivial variety $X$ with non-trivial fundamental group $\pi_1(X_{\reg})$ is not captured by its algebra of global holomorphic differential forms $H^{0}\left(X,{\Omega_X}^{[\cdot]}\right)$. Many examples supporting this fact are exposed in \cite[\S14]{GGK}, and most notably, smooth $K$-trivial threefolds with Beauville-Bogomolov decomposition of pure abelian type and algebra of global differential forms generated by the volume form (as for a Calabi-Yau threefold) are classified in \cite{OgSak}.} This paper aims at describing changes of the type of a $K$-trivial variety through a birational morphism in the extremal case, i.e., when a singular variety with Beauville-Bogomolov decomposition of purely abelian type is resolved by a Calabi-Yau manifold. We~work in the following set-up: By a Calabi-Yau manifold, we~mean a smooth simply-connected complex projective variety of dimension $n$ with trivial canonical bundle, without any global holomorphic differential form of degree $1\le i\le n-1$. Extending the terminology of \cite{OgFibSp}, we~define $n$-dimensional Calabi-Yau manifolds of type $n_0$ as follows. \begin{theorem}[{\cite{SBW}, \cite[Rem.\,1.5]{LuTaji}}] \label{thm-LuTaji} Let $X$ be a Calabi-Yau manifold of dimension~$n$. The following are equivalent: \begin{enumeratei} \item There is a nef and big divisor $D$ on $X$ such that $c_2(X)\cdot D^{n-2}=0$. \item There is an abelian variety $A$ and a finite group $G$ acting freely in codimension~$2$ on~$A$ such that $X$ is a crepant resolution of $A/G$. \end{enumeratei} If it satisfies these conditions, $X$ is called a Calabi-Yau manifold of type $n_0$. \end{theorem} Calabi-Yau threefolds of type $\textup{III}_0$ appear naturally when classifying extremal contractions of Calabi-Yau threefolds \cite{OgFibSp}, and fit in a more general circle of ideas on how the cubic intersection form and the second Chern class determine the birational geometry of a Calabi-Yau threefold (see, e.g., the work of Wilson \cite{WilKahCone}, Oguiso and Peternell \cite{OgPet}). Calabi-Yau threefolds of type $\textup{III}_0$ were classified by Oguiso, as we now recall. \begin{theorem}[\cite{III0}]\label{thm-Oguiso} There are exactly two Calabi-Yau threefolds $X_3$, $X_7$ of type $\textup{III}_0$. They are the unique crepant resolution of ${E_j}^3$ quotiented by the group generated by $j\id_3$, and of ${E_{u_7}}^3$ quotiented by the group generated by: \[ \left(\begin{array}{ccc} 0 & -8 & 7-10u_7\\ 1 & -6-2u_7 & 11-u_7\\ 0 & -1-2u_7 & 6+3u_7\\ \end{array}\right). \] where $j=e^{2i\pi/3}$, $\zeta_7=e^{2i\pi/7}$, $u_7=\zeta_7+{\zeta_7}^2+{\zeta_7}^4=\psfrac{-1+i\sqrt{7}}{2}$, and for any complex number $z\in\C\setminus\R$, we~denote by $E_z$ the elliptic curve $\C/(\Z\oplus z\Z)$. \end{theorem} Our first theorem restricts the isogeny type of $A$ in arbitrary dimension. \theo[theo-babymain]{Let $A$ be an abelian variety of dimension $n$ and $G$ be a finite group acting freely in codimension~$2$ on $A$. If~$A/G$ has a resolution that is a Calabi-Yau manifold, then $A$ is isogenous to ${E_j}^n$ or to ${E_{u_7}}^n$ and $G$ is generated by its elements that admit fixed points in $A$.} Moreover, the local geometry of $A/G$ is generally quite similar to the $3$-dimensional model (see Theorem \ref{theo-main} below). Two important consequences of this are the following results. \theo[theo-freecod3]{Let $A$ be an abelian variety and $G$ be a finite group acting freely in codimension~$3$ on $A$. Then the quotient $A/G$ has a resolution $X$ with $c_1(X)=0$ if and only if $G$ acts freely on $A$.} \theo[theo-dim4]{Let $A$ be an abelian fourfold and $G$ be a finite group acting freely in codimension~$2$ on $A$, yet not freely on A. If~$A/G$ has a resolution $X$ with $c_1(X)=0$, then there is a finite étale Galois cover $\tilde{X}$ of $X$ that is a product of the form $E\times X_3$ or $E\times X_7$, where $E$ is an elliptic curve and $X_3$, $X_7$ are the two Calabi-Yau threefolds of $\textup{III}_0$ type.} In particular, there are no new, irreducible examples arising in dimension 4. We~know of no irreducible example in dimension $n\ge 5$ either, and the discussion at the end of the introduction leads us to conjecture that there are none. The importance of quotients that have a simply-connected, Calabi-Yau resolution within the wider range of quotients that have a resolution $X$ with $c_1(X)=0$ is highlighted by a decomposition result in spirit of the Beauville-Bogomolov decomposition, see Lemma \ref{lem_bbc2}. On the way, we~make a substantial use of a necessary criterion due to Yamagishi \cite[Th.\,1.1]{Yama} for certain quotient singularities to admit crepant resolutions. This criterion can be stated as follows. \prop[prop-yama]{Let $G\subset\SL_n(\C)$ be a finite group acting freely in codimension~$1$ on~$\C^n$, and let $U\subset \C^n$ be a $G$-stable simply-connected analytic neighborhood of \hbox{$0\in\C^n$}. If~the singularity $U/G$ admits a crepant resolution $X_G$, then the group $G$ is generated by junior elements.} It is worth noting that this criterion is a weaker version of the following conjecture by Ito and Reid \cite[4.5]{ItoReid}. \begin{conjecture} Let $G\subset\SL_n(\C)$ be a finite group acting freely in codimension~$1$ on~$\C^n$, and let $U\subset \C^n$ be a $G$-stable simply-connected analytic neighborhood of $0\in\C^n$. If~the singularity $U/G$ admits a crepant resolution $X_G$, then every maximal cyclic subgroup of $G$ is generated by a junior element. \end{conjecture} On our way to proving Theorem \ref{theo-main} below, we~look for more local arguments on whether particular, explicit finite quotient singularities admit a crepant resolution. This leads us to find a counterexample to the conjecture \cite[4.5]{ItoReid} above, which is strong evidence that the necessary criterion \cite[Th.\,1.1]{Yama}, albeit far from sufficient (see, e.g., \cite[Ex.\,3.11]{Sato}), would be hard to improve. Our counterexample features a faithful representation of the group $\SL_2(\F_3)\simeq Q_8\rtimes\Z_3$ into $\SL_6(\C)$, and is presented in more detail in Remark \ref{rem-macaulay}. \smallskip The structure of the paper is as follows. Sections \ref{sec-McKay} to \ref{sec-pftheomain} build up to the proof of the main technical result. \theo[theo-main]{Let $A$ be an abelian variety of dimension $n$ and $G$ be a finite group acting freely in codimension~$2$ on $A$. If~$A/G$ has a resolution that is a Calabi-Yau manifold, then \begin{enumerate} \item $A$ is isogenous to ${E_j}^n$ or to ${E_{u_7}}^n$, and $G$ is generated by its elements that admit fixed points in $A$. \item For every translated abelian subvariety $W\subset A$, there is a integer $k\ge 0$ such that the \emph{pointwise stabilizer} \[ \pstab(W):=\{g\in G\mid \forall w\in W,\, g(w)=w\} \] is isomorphic to ${\Z_3}^k$ if $A$ is isogenous to ${E_j}^n$, or to ${\Z_7}^k$ if $A$ is isogenous to ${E_{u_7}}^n$. \item For every translated abelian subvariety $W\subset A$, if $ \pstab(W)$ is isomorphic to \begin{itemize} \item ${\Z_3}^k$ for some $k\ge 1$, then there are $k$ generators of it such that their matrices are similar to $\diag(\1_{n-3},j,j,j)$, and the $j$-eigenspaces of these matrices are in direct sum. \item ${\Z_7}^k$ for some $k\ge 1$, then there are $k$ generators of it such that their matrices are similar to $\diag(\1_{n-3},\zeta_7,{\zeta_7}^2,{\zeta_7}^4)$, and all eigenspaces of these matrices with eigenvalues other than $1$ are in direct sum. \end{itemize} \end{enumerate} } Our starting point in Section \ref{sec-McKay} is to recall the necessary condition \cite[Th.\,1.1]{Yama}, phrased above as Proposition \ref{prop-yama} for a local quotient singularity to admit a crepant resolution. This criterion sheds light on the importance of the so-called junior elements of $\SL_n(\C)$ when studying the birational geometry of finite quotient singularities in dimension $n$. A junior element is a matrix $M\in\SL_n(\C)$ of finite order $d$, whose eigenvalues $(e^{2i\pi a_k/d})_{1\le k\le n}$ with $0\le a_k\le d-1$ satisfy $\sum a_k = d$.\enlargethispage{\baselineskip} Matrices inducing actions on abelian varieties satisfy a rationality requirement \cite[Prop.\,1.2.3]{BirkLang}, which translates into arithmetic constraints on their characteristic polynomial. These constraints allow us to classify matrices of junior elements~$g$ acting on $n$-dimensional abelian varieties up to similarity: In Section \ref{sec-JuniorClassification}, we~prove that if a junior element~$g$ acts on an abelian variety in a way that the generated group $\grp{g}$ acts freely in codimension~$2$, then the matrix of~$g$ is of one of twelve possible types (see Proposition \ref{prop-junclass}). In particular, the order of~$g$ and the number of non-trivial eigenvalues of~$g$ are bounded independently of the dimension $n$. \smallskip The next step is to show that ten out of the twelve types of junior elements can not belong to $G$, for a mix of local and global reasons. The proof spreads throughout Sections \ref{sec-CyclicPointwiseStabilizer}, \ref{sec-RuleOutMostJuniors}, \ref{sec-codim5} and \ref{sec-codim6}. Let us sketch the idea of the argument in the simplest case, namely if~$g$ is a junior element of composite order other than $6$, with at most four non-trivial eigenvalues. If~such a junior element~$g$ belongs to $G$, then some non-trivial power~$g^{\alpha}$ is not junior, and has a larger fixed locus in $A$. Fix an irreducible component~$W$ of that larger fixed locus that is not in the fixed locus of~$g$: the pointwise stabilizer $\pstab(W)\subset G$ does not contain~$g$, but the power $g^{\alpha}$. Now, as $W$ has codimension less than $4$, Section \ref{sec-CyclicPointwiseStabilizer} shows that $\pstab(W)$ is cyclic generated by one junior element~$h$, and thus, up to possibly replacing~$h$ by another junior generator of $\fix(W)$, one has $g^{\alpha}=h^{\alpha}$. For well-chosen $\alpha$, this is enough to yield $g=h$, and a~contradiction. This idea excludes seven out of the twelve types of junior elements (see Section~\ref{subsec-RuleOutSevenJuniors}). The three types of junior elements of order 6 are excluded by technical variations in the next sections. Ruling them out works along with classifying pointwise stabilizers in higher codimension: In codimension~$4$ , Section \ref{sec-CyclicPointwiseStabilizer} establishes cyclicity of the pointwise stabilizers and Section \ref{sec-RuleOutMostJuniors} deduces that junior elements with four non-trivial eigenvalues do not exist; in codimension~5 (Section \ref{sec-codim5}), we~first prove that junior elements with five non-trivial eigenvalues do not exist (Section \ref{subsec-ruleout65}), then deduce cyclicity of the pointwise stabilizers (Section \ref{subsec-EigenspaceDirectSumCod5}). In codimension~6 (Section \ref{sec-codim6}), we~first classify pointwise stabilizers which do not contain junior elements of type $\diag(\1_{n-6},\omega,\omega,\omega,\omega,\omega,\omega)$: they are isomorphic to $\Z_3,\Z_7,\Z_3\times\Z_3,\Z_7\times\Z_7$, or $\SL_2(\F_3)$ (Section \ref{subsec-pstabcod6}). We~use this partial classification to rule out junior elements with six non-trivial eigenvalues (Section \ref{subsec-ruleout66}), and we then finally refine the study of pointwise stabilizers in codimension~$6$ by ruling out $\SL_2(\F_3)$ (Section \ref{subsec-JuniorsCommute}). There finally remain two types of possible junior elements, which are those already appearing in dimension $3$ in \cite{III0}: $\diag(\1_{n-3},j,j,j)$ and $\diag(\1_{n-3},\zeta_7,{\zeta_7}^2,{\zeta_7}^4)$. This description of pointwise stabilizers in codimension up to 6 implies that any two junior elements admitting a common fixed point commute. Together with a simple argument about the isogeny type of $A$ (see Section \ref{sec-MakeDisjoint}), it~concludes the proof of Theorem \ref{theo-main}. In fact, the idea that the existence of certain automorphisms on an abelian variety determines the isomorphism type of some special abelian subvarieties is general (\cite{Shimura}), and it applies crucially throughout this paper, starting in Section~\ref{sec-CyclicPointwiseStabilizer}. From there, it~is not so surprising that we are able to determine the isogeny type of~$A$, interpreting the fact that $A/G$ admits a Calabi-Yau resolution as an irreducibility property of the $G$-equivariant Poincaré decomposition of $A$. \smallskip Under the additional assumption that the group $G$ is abelian, Theorem \ref{theo-main} and the results of Section \ref{sec-MakeDisjoint} suffice to generalize Theorem \ref{theo-dim4} to higher dimensions, i.e., to the statement that, if $A$ is an abelian variety of dimension $n$ and $G$ is a finite group acting freely in codimension~$2$ on $A$ such that $A/G$ admits a Calabi-Yau resolution~$X$, then $n=3$ and $X$ is $X_3$ or $X_7$. Also note that $G$ is abelian if and only if any two junior elements $g,h$ of $G$ commute, which by our results can be checked via their matrices acting on a vector space $V$ of dimension $3$, $4$, $5$, or $6$. Standard finite group theory allows us to explicitly bound the order of $\grp{g,h}$ depending on this dimension and the isogeny type of $A$. If~the dimension is $3$ or 4, the bounds are reasonable enough to launch a computer-assisted search through all possible abstract groups $\grp{g,h}$. Among these, the only groups which, in a faithful $3$ or 4-dimensional representation, are generated by two junior elements of the same type, are $\Z_3$, $\Z_7$, and the finite simple group $\SL_3(\F_2)$ of order 168. But a geometric argument on fixed loci excludes $\SL_3(\F_2)$, whence the wished contradiction. This reproves the classification of \cite{III0} in dimension 3, and settles Theorem \ref{theo-dim4}. When $V$ has dimension 5 or $6$, we~could also bound the order of $\grp{g,h}$ explicitly. For example, we~could consider the image of the faithful representation $M\oplus\overline{M}$ in $\SL_{2\dim(V)}(\Q)$, and use the classification of irreducible maximal finite integral matrix groups in dimension less than 12 by V.\,Felsch, G.\,Nebe, W.\,Plesken, and B.\,Souvignier to obtain a bound on the order of $\grp{g,h}$. But the bounds obtained in this way are too large for the \texttt{SmallGroup} library. One needs to better understand the arising matrix groups of larger order, and build a reasonably smaller finite list of possibilities for the abstract group $\grp{g,h}$. It will then remain to figure out geometric ways for ruling out those potential groups in the list other than $\Z_3$, $\Z_7$, $\Z_3\times\Z_3$, and $\Z_7\times\Z_7$. \smallskip Some of our proofs resort to computer-searches among all finite groups of certain fixed orders (relying on the \texttt{SmallGroup} library of \texttt{GAP}). The computer-assisted results used in Section \ref{subsec-GroupThPointwiseStabilizer} were actually originally proved by hand using elementary representation theory and Sylow theory. Such arguments being standard in finite group theory, we~chose to keep their exposition concise for the sake of readability, and preferred invoking computer-checked facts as black boxes when needed. This approach also has the advantage of better separating abstract group-theoretic arguments on~$G$ from properties of the particular representation $G\hto\GL(H^0(T_A))$. All programs used are available in the appendix. \subsubsection*{Acknowledgements} I am grateful to my adviser Andreas Höring for fruitful discussions. This paper has greatly benefit from an anonymous referee's report, and I want to thank them for their time, care, commitment, and for bringing the highly relevant results of \cite{Yama} to my attention. Finally, I thank Julia Schneider for suggesting to state and prove Theorem \ref{theo-freecod3}, and Stéphane Druel, for proposing the current, better statement of Theorem \ref{theo-freecod3}. \section{Some results in McKay correspondence}\label{sec-McKay} \defi{Let~$g$ be a matrix in $\GL_n(\C)$. Assume that it has finite order $d$. Since $g^d = \id$,~$g$ is diagonalizable and has eigenvalues of the form $e^{2i\pi a_k/d}$, for integers $0\le a_1\le\ldots\le a_n\le d-1$. (We allow redundancies.) We define the \emph{ranked vector of eigenvalues} of~$g$ as the tuple $(e^{2i\pi a_k/d})_{1\le k\le n}$. The \emph{age} of~$g$ is set to be the number $\psfrac{a_1+\ldots+a_n}{d}$. If~it equals 1, we~say that~$g$ is \emph{junior}.} \defi[def-juniorab]{If $A$ is an abelian variety of dimension $n$ and $g\in\Aut(A)$ has finite order, then~$g$ can be written as: \[ g:[z]\in A\mto [M(g)z+T(g)]\in A, \] where $M(g)$ is a matrix of finite order in $\GL_n(\C)$, $T(g)$ a vector in $\C^n$. If~$g$ fixes any point $a$ of $A$, it~can be represented locally in a neighborhood of $a$ by its matrix $M(g)$. Hence, it~makes sense to say that the automorphism~$g$ is \emph{junior} if~$g$ fixes at least one point in $A$ and the matrix $M(g)$ is junior.} \rem{Note that if $g\in\Aut(A)$ admits a fixed point, then $\grp{g}$ contains no translation, so~$g$ and its matrix $M(g)$ have the same order.} Junior elements play a key role in the study of finite quotient singularities, as the following theorem emphasizes. \begin{theorem}[\cite{ItoReid}] \label{thm-IR} Let $G$ be a finite subgroup of $\SL_n(\C)$. Suppose that the finite Gorenstein quotient $\C^n/G$ has a minimal model $X$. Then there is a natural one-to-one correspondence between conjugacy classes of junior elements in $G$ and prime exceptional divisors in $X$. \end{theorem} \rem{Note that such a minimal model $X$ always exists as a relative minimal model of a resolution $\tilde{X}\to\C^n/G$, by \cite[1.30.6]{Singbook}.} Quotient singularities are $\Q$-factorial, so they can not be resolved by small birational morphisms. This yields a simple corollary of the theorem. \begin{corollary}[\cite{ItoReid}] \label{cor-IR} Let $G$ be a finite subgroup of $\SL_n(\C)$. If~the Gorenstein quotient singularity $\C^n/G$ admits a crepant resolution, then there is a junior element $g\in G$. \end{corollary} In fact, \cite[\S4.5]{ItoReid} conjectures that under the same hypotheses, if the singularity $\C^n/G$ admits a crepant resolution, then any maximal cyclic subgroup of $G$ contains a junior element. A counterexample to this conjecture is however presented in Remark~\ref{rem-macaulay}. We will also use a slightly stronger version of Corollary \ref{cor-IR} for quotients by finite subgroups of $\GL_n(\C)$. The formulation is inspired by \cite[Th.\,2.3]{MorrisonStevens}. \begin{proposition}[\cite{Reid,RSB,Tai}] \label{prop-Reid} Let $G$ be a finite subgroup of $\GL_n(\C)$. If~the Gorenstein quotient singularity $\C^n/G$ admits a crepant resolution, then there is a junior element $g\in G$. \end{proposition} \subsection{A local necessary criterion} The following result, due to \cite{Yama}, provides a valuable necessary criterion for the existence of a crepant resolution to an affine quotient singularity, in the spirit of \cite{ItoReid}. \prop[prop-genjun]{Let $G\subset\SL_n(\C)$ be a finite group acting freely in codimension~$1$ on $\C^n$, If the affine quotient singularity $\C^n/G$ admits a crepant resolution, then the group $G$ is generated by junior elements.} The following corollary allows us to apply this proposition to some finite groups which are a~priori contained in $\GL_n(\C)$ and not $\SL_n(\C)$. We~recall that a \emph{$K$-trivial variety} is a smooth projective variety $X$ with $K_X\sim\O_X$. \cor[cor-pstabjun]{Let $G$ be a finite group acting freely in codimension~$1$ on an abelian variety $A$. Suppose that $A/G$ has a $K$-trivial resolution $X$. Then for every point $a\in A$, the stabilizer $\Stab(a):=\{g\in G\mid g(a)=a\}$ is generated by junior elements.} \demo{Since $X$ is $K$-trivial, we~have $h^n(X,\O_X)=1$, where $n$ denotes the dimension of $X$. Since $A/G$ has rational singularities, this yields $h^n(A/G,\O_{A/G})=1$, hence there exists a non-zero $G$-invariant element in $H^n(A,\O_A)$. Since $A$ is an abelian variety, the action of $G$ on $H^n(A,\O_A)=\bigwedge^n H^1(A,\O_A)$ is given by the dual of the determinant representation $\det M: G\to \GL_n(\C)\to \C^*$. The fact that there is a non-zero invariant element shows that $\det M$ is trivial, i.e., $M(G)\subset \SL_n(\C)$. Fix $a\in A$. Note that, near the point $a$, the variety $A/G$ is locally analytically isomorphic to the affine quotient singularity $\C^n/\Stab(a)$ near the origin $0$ -- with the action by $M:\Stab(a)\to\SL_n(\C)$. Since $A/G$ admits a crepant resolution, the affine quotient singularity $\C^n/\Stab(a)$ admits a crepant resolution too. Applying Proposition \ref{prop-genjun} yields that the subgroup $M(\Stab(a))<\SL_n(\C)$ is generated by junior elements, and this concludes. } \subsection{A global result along the same lines} We also prove a global result along the same lines as Proposition \ref{prop-genjun}. \lem[lem-makeitglob]{Let $G$ be a finite group acting freely in codimension~$1$ on an abelian variety $A$. Suppose that $A/G$ has a resolution $X_G$ with $c_1(X_G)=0$ and $\pi_1(X_G)=\{1\}$ Then $G$ is generated by its elements admitting fixed points in $A$.} \demo{Let $G_0\triangleleft G$ be the normal subgroup of $G$ generated by elements admitting fixed points. We~want to prove that $G_0 = G$. We~have a commutative diagram given by the fiber product: \[ \xymatrix{ X_0\ar[d]_{\eps_0}\ar[r]^{\tilde{q}} & X_G\ar[d]^{\eps_G} \\ A/G_0\ar[r]^{q} & A/G } \] By definition of $G_0$, for every $a\in A$, the stabilizers of $a$ in $G$ and $G_0$ coincide. Hence, $q$ is étale, and $\tilde{q}$ is étale too by base change. But $X_G$ is simply-connected and~$X_0$ is connected, so $\deg(\tilde{q})=1$ and $G_0 = G$.} \rem{If $G$ is a finite group acting freely in codimension~$1$ on an abelian variety $A$ so that $A/G$ has a resolution $X$ with $c_1(X)=0$ and $\pi_1(X)=\{1\}$, then~$G$ may still contain elements that admit no fixed point. Without loss of generality, we~can assume that $G$ contains no translation, up to replacing $A$ by an isogenous abelian variety, but that is the best we can do.} \Subsection{A reduction result inspired by the Beauville-Bogomolov decomposition} We conclude this section with a reduction result in the spirit of the Beauville--Bogomolov decomposition. We~reiterate that we include simply connectedness in the definition of a Calabi--Yau variety. \lem[lem_bbc2]{Let $X$ be a smooth projective variety with $c_1(X)=0$. Assume that~$X$ is a resolution of a quotient $A/G$, where $A$ is an abelian variety, and $G$ is a finite group acting freely in codimension~$2$ on $A$. Then, there is a finite étale Galois cover $p:\tilde{X}\to X$ such that \[ \tilde{X}= B\times\prod_{i=1}^r Y_i, \] where $B$ is an abelian variety, and each $Y_i$ is a Calabi-Yau variety of dimension at least $3$ that resolves a quotient $B_i/H_i$, where $B_i$ is an abelian variety and $H_i$ is a finite group acting freely in codimension~$2$ on $B_i$.} \demo{Let $n$ denote the dimension of $X$. Pulling back an ample divisor on $A/G$ to $X$, we~have a nef and big Cartier divisor $D$ on $X$ such that $c_2(X)\cdot D^{n-2}=0$. By~the Beauville--Bogomolov decomposition theorem \cite{Beauv84}, we~can take a finite étale Galois cover $p:\tilde{X}\to X$ that decomposes as \[ \tilde{X}= B\times\prod_{i=1}^r Y_i\times \prod_{j=1}^s Z_j, \] where $B$ is an abelian variety, each $Y_i$ is a Calabi-Yau variety of dimension at least 3, and each $Z_j$ is an irreducible holomorphic symplectic variety. Since $h^1(Y_i,\O_{Y_i})=0$ for each $i$ and $h^1(Z_j,\O_{Z_j})=0$ for each $j$, we~can use \cite[Exer.\,III.12.6]{HartBook} to write \[ p^*D=p_B^*D_B+\sum_{i=1}^r p_i^*L_i+\sum_{j=1}^s q_j^*M_j, \] where $D_B$, $L_i$, $M_j$ are divisors on $B$, $Y_i$, $Z_j$ respectively, and $p_B,p_i,q_j$ are the projections onto $B$, $Y_i$, $Z_j$. By the projection formula and since $D^n > 0$, it~is easy to check that each of the divisors $D_B$, $L_i$, $M_j$ is nef and big. As the tangent bundle of $\tilde{X}$ decomposes into a direct sum of the tangent bundles of $B$, of the $Y_i$, and of the $Z_j$, we~also note that \[ c_2(\tilde{X}) = \sum_{i=1}^r p_i^*c_2(Y_i) + \sum_{j=1}^s q_j^*c_2(Z_j). \] Since the second Chern class of a $K$-trivial variety of dimension $m$ has non-negative intersection number with the product of $(m-2)$ nef divisors \cite{Miyaoka}, and since $c_2(X)\cdot D^{n-2}=0$, we~obtain $c_2(Y_i)\cdot L_i^{\dim Y_i - 2} = 0$ for all $i$ and $c_2(Z_j)\cdot M_j^{\dim Z_j - 2} = 0$ for all~$j$. By \cite[Rem.\,1.5]{LuTaji}, \cite{SBW}, this shows that each $Y_i$ resolves a quotient $B_i/H_i$, where~$B_i$ is an abelian variety and $H_i$ is a finite group acting freely in codimension~$2$ on $B_i$, and that each $Z_j$ resolves a quotient $C_j/K_j$, where $C_j$ is an abelian variety and $K_j$ is a finite group acting freely in codimension~$2$ on $C_j$. To conclude, we~note that each of the finite quotients $C_j/K_j$ has a symplectic resolution $Z_j$ and is smooth in codimension~$2$. Hence \cite[Th., Cor.\,1]{NamiSymplectic} applies, and $C_j/K_j$ has terminal singularities. As a terminal $\Q$-factorial variety that is $K$-trivial, it~cannot have a $K$-trivial resolution. So there are no irreducible holomorphic symplectic factors $Z_j$ in $\tilde{X}$, and this concludes this proof.} \section{The twelve types of junior elements on an abelian variety}\label{sec-JuniorClassification} Section \ref{sec-McKay} just shows that, if we want a finite singular quotient of an abelian variety $A/G$ to have a resolution $X$ with $c_1(X)=0$, the group $G$ must contain some junior elements. The fact that in our set-up, $G$ must also act freely in codimension~$2$ on $A$ is restrictive enough that there are only twelve possibilities for the ranked vector of eigenvalues of a junior element $g\in G$. \prop[prop-junclass]{Let $A$ be an abelian variety of dimension $n$, and $g\in\Aut(A)$ be a junior element such that $\grp{g}$ acts freely in codimension~$2$. Then the order $d$ of~$g$ and the ranked vector of eigenvalues of~$g$ are in one of the twelve columns of Table \ref{tab-propjunclass}.} \begin{table}[htb] \caption{Possible ranked vectors of eigenvalues for junior elements in $G$. For $d\in\N$, we~denote $\zeta_d =e^{2i\pi/d}$, and in particular $j=e^{2i\pi/3}$ and $\omega=e^{2i\pi/6}$. For $k\in\N$, $\1_k$ refers to a sequence of $k$ times the symbol $1$ in a row.} \label{tab-propjunclass} \renewcommand{\arraystretch}{1.3} \begin{adjustbox}{width=\columnwidth,center} $\begin{array}{|c|c|c|c|} \hline d & 3 & 4 & 6 \\ \hline (e^{2i\pi a_k/d}) &\left(\1_{n-3},j,j,j\right) &\left(\1_{n-4},i,i,i,i\right) &\left(\1_{n-4},\omega,\omega,\omega,-1\right)\\ \hline \hline d & 6 & 6 & 7 \\ \hline (e^{2i\pi a_k/d}) &\left(\1_{n-5},\omega,\omega,\omega,\omega,j\right) &\left(\1_{n-6},\omega,\omega,\omega,\omega,\omega,\omega\right) & \left(\1_{n-3},\zeta_7,{\zeta_7}^2,{\zeta_7}^4\right)\\ \hline \hline d & 8 & 12 & 15 \\ \hline (e^{2i\pi a_k/d}) & \left(\1_{n-4},\zeta_8,\zeta_8,\zeta_8^3,\zeta_8^3\right) & \left(\1_{n-4},\zeta_{12},\zeta_{12},\zeta_{12}^5,\zeta_{12}^5\right) &\left(\1_{n-4},\zeta_{15},\zeta_{15}^2,\zeta_{15}^4,\zeta_{15}^8\right)\\ \hline \hline d & 16 & 20 & 24\\ \hline (e^{2i\pi a_k/d}) & \left(\1_{n-4},\zeta_{16},\zeta_{16}^3,\zeta_{16}^5,\zeta_{16}^7\right) & \left(\1_{n-4},\zeta_{20},\zeta_{20}^3,\zeta_{20}^7,\zeta_{20}^9\right) & \left(\1_{n-4},\zeta_{24},\zeta_{24}^5,\zeta_{24}^7,\zeta_{24}^{11}\right)\\ \hline \end{array}$ \end{adjustbox} \end{table} The proof goes by elementary arithmetic and meticulous case disjunctions. The following terminology should simplify the exposition. \defi{A \emph{multiset} $\bA$ is the data of a set $A$ and a function $m: A\to \Z_{>0}$, called the \emph{multiplicity function}. Intuitively, a multiset is like a set where elements are allowed to appear more than once. If a multiset $\bA=(A,m)$ is \emph{finite}, i.e., its underlying set $A=\{a_1,\ldots,a_k\}$ is finite, we~may write $A$ in the following form: \[ \{\{\underset{m(a_1)\mbox{\footnotesize{ times}}}{\underbrace{a_1,\ldots,a_1}},\ldots,\underset{m(a_k)\mbox{\footnotesize{ times}}}{\underbrace{a_k,\ldots,a_k}}\}\}. \] Double-braces are used to avoid confusion between the multiset and the underlying set. Let $\bA=(A,m)$ be a finite multiset. \begin{itemize} \item If $\alpha\in\Z_{>0}$, we~denote by $\bA^{*\alpha}$ the multiset $(A, \alpha m)$. \item If $A$ a subset of $\Q$, and $p,q$ are rational numbers, with $q\ne 0$, we~denote by $p+q\bA$ the multiset $(p+qA, m')$, where the multiplicity function $m':p+qA\to\Z_{>0}$ is obtained composing the affine base change $x\in p+qA\mto \psfrac{x-p}{q}\in A$ with $m$. \item The \emph{cardinal} of $\bA$ is: \[ |\bA|:=\displaystyle\sum_{a\in\bA}m(a). \] \item More generally, if $f:A\to \Q$ is a function, we~define: \[ \displaystyle\sum_{a\in \bA} f(a):=\displaystyle\sum_{a\in{A}} m(a)f(a). \] \item If $\bA=(A,m)$ and $\bB=(B,n)$ are two multisets, we~define their \emph{union}: \[ \bA\cup\bB := (A\cup B, \1_Am+\1_Bn), \] where $\1_A,\,\1_B$ are the indicator functions of $A$ and $B$. \end{itemize} } \nota{For $d\in\N$, we~denote by $\Phi_d$ the $d$-th cyclotomic polynomial, and by $\phi(d)$ the degree of $\Phi_d$. In other terms, $\phi$ is the Euler indicator function. For integers $a,b$, the greatest common divisor of $a$ and $b$ is denoted $a\wedge b$.} We establish a sequence of three useful lemmas. \lem[lem-inteq]{Let $u$ be a positive integer strictly greater than $2$. Then we have: \begin{gather*} \Bigl(2\nmid u\mbox{ and }\frac{\phi(u)^2}{u}\le 8\Bigr) \mbox{ or } \Bigl(2\mid u\mbox{ and }\frac{\phi(u)^2}{u}\le 4\Bigr) \\ \iff\;u\in\lint 3, 10\rint\cup \{12,14,15,16,18,20,21,24,30,36,42\}. \end{gather*} } \demo{Write $u=p_1^{\alpha_1}\cdot p_2^{\alpha_2}\cdots p_k^{\alpha_k}$, where $p_1<\ldots 0$, or the volume form of $Y_J/M(G)$ if $\dim Y_I = 0$ (in which case $\dim Y_J > 0$). This yields an $M(G)$-invariant non-zero global holomorphic form of positive degree $y$ on $A$. Note that the sections of $\Omega_{A}^{\cdot}$ are invariant by translations of $A$, so that we in fact have a $G$-invariant non-zero global holomorphic $y$-form on $A$. It pulls back to $X$, which is a Calabi-Yau variety. Hence $y=n$, and either $A\sim {E_j}^{n}$ or $A\sim {E_{u_7}}^{n}$. The order of junior elements generating $G$ is given accordingly by Propositions \ref{prop-singab}, \ref{prop-globfailjun}.} \section{Junior elements and pointwise stabilizers in codimension 5}\label{sec-codim5} In this section, we~extend the results of Sections \ref{sec-CyclicPointwiseStabilizer} and \ref{sec-RuleOutMostJuniors} to codimension $k=5$. In the first subsection, we~exclude the one type of junior element with exactly five non-trivial eigenvalues. In the second subsection, we~prove the following result. \prop[prop-fixor5]{Let $A$ be an abelian variety on which a finite group $G$ acts freely in codimension~$2$. Suppose that $A/G$ has a $K$-trivial resolution. Let $W$ be a translated abelian subvariety of codimension $k\le 5$ in $A$ such that $\pstab(W)\ne\{1\}$. Then $ \pstab(W)$ is a cyclic group, generated by one junior element~$g$ of order~$3$ or $7$.} \Subsection{Ruling out junior elements of order 6 with five non-trivial eigenvalues}\label{subsec-ruleout65} \prop[prop-jun6cod5]{Let $A$ be an abelian variety, $G$ a group acting freely in codimension~$2$ on $A$ such that $A/G$ has $K$-trivial resolution. Then there is no junior element of $G$ whose matrix is similar to $\diag(\1_{n-5}, \omega,\omega,\omega,\omega,j)$.} \demo{Suppose by contradiction that there is an element $g\in G$ such that $g(0)=0$ and, in some coordinates, \[ M(g) = \diag(\1_{n-5}, \omega,\omega,\omega,\omega,j). \] Then there is an abelian subvariety $W$ of codimension~$4$ in $A$ which is pointwise fixed by $g^3$. By~Proposition \ref{prop-cyclicincod4}, $\pstab(W)$ is cyclic, generated by one junior element~$h$. As $g^3\in\grp{h}$,~$h$ has even order. However, by Propositions \ref{prop-globfailjun} and \ref{prop-ruleout6cod4}, it~must have order~$3$ or $7$, contradiction!} \subsection{The pointwise stabilizer for loci of codimension 5}\label{subsec-EigenspaceDirectSumCod5} For proving Proposition~\ref{prop-fixor5}, it~is enough to establish the following result. \prop[prop-fixor5local]{Let $B$ be an abelian fivefold isogenous to either ${E_j}^5$ or ${E_{u_7}}^5$, and let $p=3$ in the first case, $p=7$ in the second case. Let $F$ be a finite subgroup of $\Aut(B,0)$ generated by junior elements of order $p$, satisfying the following condition: For any non-trivial subgroup $F_0 (n-4)|\grp{g,h}|, \] by Lemma \ref{lem-class7dim4}, the inequality being strict since $\grp{g,h}$ contains at least one junior element of order $7$. Hence, ${\bf 1}$ has multiplicity at least $n-3$ as a subrepresentation of~$M$, i.e., $E_{M(g)}(1)\cap E_{M(h)}(1)$ has codimension at most $3$ in $H^0(T_A)$.} We now prove an auxiliary lemma for Proposition \ref{prop-dim4gen3}. \lem[lem-gh3group]{Let $A$ be an abelian variety isogenous to ${E_j}^n$. Let $g,h\in\Aut(A)$ be two junior elements of order~$3$ such that $\grp{g,h}$ contains no translation and no non-junior element fixing points, and $E_{M(g)}(1)\cap E_{M(h)}(1)$ has codimension~$4$ in $H^0(T_A)$. Then each non-trivial element of $\grp{g,h}$ has order~$3$.} \demo{Up to conjugating both~$g$ and~$h$ by the same translation, we~can assume $g(0)=0$. Note that the linear automorphism $M(h)$ also fixes $0$, and let $W$ be the abelian variety fixed by $\grp{g,M(h)}$. It is a subvariety of codimension~$4$ of $A$. Let $G:= \grp{g,h}$. Let $G_W$ be the subgroup of $G$ defined by Lemma \ref{lem-generalGW}. Again, we~have $G_W=\grp{g,h}$. Consider the faithful representation $\rho$ of $\grp{g,h}$ of rank four given by Lemma \ref{lem-generalGW}, with character $\chi$. Let $k\in\grp{g,h}$. If~$k$ has a fixed point in $A$, then $k$ is junior of order~$3$. Otherwise, by Lemma \ref{lem-generalGW} (5), $1$ is an eigenvalue of $\rho(k)$. In that case, since the intersection $E_{\rho(g)}(j)\cap E_{\rho(h)}(j)$ has dimension $2$, one of the roots of unity $1,j,j^2$ must appear as an eigenvalue of multiplicity $2$ for $\rho(k)$. By~Lemma \ref{lem-blrat}, \cite[Prop.\,2.4]{Wash}, and since $\rho(k)$ has determinant one, the characteristic polynomial of $\rho(k)$, defined over $\Q[j]$, is one of the following\vspace*{-3pt} \[ {\Phi_1}^4,\; {\Phi_1}^2{\Phi_2}^2,\;{\Phi_1}^2\Phi_3,\;{\Phi_1}^2\Phi_4,\;{\Phi_1}^2\Phi_6,\;\Phi_1(X)(X-j)^3,\; {\Phi_1}(X)(X-j^2)^3. \] So the order of $k$ is $1$, $3$, or an even number. To conclude, it~is enough to show that no element of $\grp{g,h}$ has order~$2$. We~prove it by contradiction: Suppose that $k\in \grp{g,h}$ is such that $\rho(k)$ is similar to $\diag(1,1,-1,-1)$. As the eigenspace $E_{\rho(g)}(j)$ has codimension~1, the matrix $\rho(gk)$ has $j$ and $-j$ as eigenvalues. In particular, it~is not junior, and thus it fixes no point. But its characteristic polynomial is not one of the polynomials listed above either, a~contradiction.} \demode{Proposition \ref{prop-dim4gen3}}{Arguing by contradiction, we~assume that the subspace $E_{M(g)}(1)\cap E_{M(h)}(1)$ has codimension~$4$ in $H^0(T_A)$. By~Lemma \ref{lem-gh3group}, the non-trivial four-by-four matrices in the group $\rho(\grp{g,h})$ are each similar to one of the following: \begin{equation}\label{eq-type3} \diag(1,j,j,j),\;\diag(1,j^2,j^2,j^2),\mbox{ or }\diag(1,1,j,j^2). \end{equation} Note in particular that $\diag(j,j,j^2,j^2)$ is not an option (since it does not have $1$ as an eigenvalue, it~would come from an element $k\in\grp{g,h}$ which fixes a point, and yet it is not junior, a~contradiction). As $\grp{g,h}$ is a $3$-group, we~can set $k\in Z(\grp{g,h})$ to be a central element of order~$3$. Since $E_{M(g)}(1)\cap E_{M(h)}(1)$ has codimension~$4$ in $H^0(T_A)$, we~know that $\grp{g}\cap\grp{h}=\{1\}$. Thus, up to renaming~$g$ and~$h$ into each other, we~can assume $k\not\in\grp{g}$. If $\rho(k)$ is similar to $\diag(1,j,j,j)$ or $\diag(1,j^2,j^2,j^2)$, then $\rho(gk)$ or $\rho(g^2k)$, respectively, has no $1$ as an eigenvalue, which contradicts (\ref{eq-type3}). Hence, $\rho(k)$ is similar to $\diag(1,1,j,j^2)$. Note that $E_{\rho(g)}(j)\cap E_{\rho(h)}(j)$ has dimension $2$, and that any element of $\rho(\grp{g,h})$ must be a homothety on this 2-dimensional subspace. Given the matrix $\rho(k)$, this means that $E_{\rho(g)}(j)\cap E_{\rho(h)}(j)=E_{\rho(k)}(1)$. Again, either $\rho(gk)$ or $\rho(g^2k)$ has no $1$ as an eigenvalue, which contradicts (\ref{eq-type3}). } %\newpage \section*{Appendix} \Subsection*{Groups of order dividing 240 with an automorphism of order 7}\vspace*{-.5\baselineskip} \bgroup\Smaller \lstinputlisting[language = GAP, firstline=3]{gachet_code/Dividing240GroupWithAutOrder7.txt} \egroup \Subsection*{Representations of $\Z_3\rtimes\Z_8$} \bgroup\Smaller \lstinputlisting[language = GAP, firstline=3]{gachet_code/Z3rtimesZ8reps.txt} \egroup \enlargethispage{-\baselineskip}\pagebreak[2] \Subsection*{Proposition \ref{prop-2sylowiscyc}: Five candidates for $F$} \bgroup\Smaller \lstinputlisting[language = GAP, firstline=3]{gachet_code/Q8orQ16.txt} \egroup \Subsection*{Proposition \ref{prop-2sylowiscyc}: Two candidates generated by elements of the right order} \bgroup\Smaller \lstinputlisting[language = GAP, firstline=3]{gachet_code/fiveoptionsgenerators.txt} \egroup \Subsection*{Proposition \ref{prop-2sylowiscyc}: None admitting the right representation} \bgroup\Smaller \lstinputlisting[language = GAP, firstline=3]{gachet_code/twooptionsreps.txt} \egroup \Subsection*{Pointwise stabilizers in codimension 6 as in Subsection \ref{subsec-pstabcod6}} \bgroup\Smaller \lstinputlisting[language = GAP, firstline=3]{gachet_code/PointwiseStab6.txt} \egroup \Subsection*{Groups of order dividing 168 as in Lemma \ref{lem-sl32appears}} \bgroup\Smaller \lstinputlisting[language = GAP, firstline=3]{gachet_code/168group.txt} \vspace*{\baselineskip} \egroup \backmatter \bibliographystyle{jepplain+eid} \bibliography{gachet} \end{document}