Compactifications of moduli of abelian varieties: an introduction (Q2918464)

In the paper under review, the author surveys the various approaches to the compactification of the moduli space of polarized abelian varieties. More precisely, let \(g\in\mathbb{N}\) is such that \(g\geq 2\), and fix \(d\in\mathbb{N}^{*}\). If \(k\) is an algebraically closed field and \(A\) is an abelian variety over \(k\), any line bundle \(L\) on \(A\) induces a morphism \(\lambda_{L}:A\longrightarrow A^{t}\) (where \(A^{t}\) denotes the dual abelian variety of \(A\)) defined by \(\lambda_{L}(x):=t_{x}^{*}L\otimes L^{-1}\), where \(t_{x}:A\longrightarrow A\) is the translation by \(x\).NEWLINENEWLINEA polarization of degree \(d\) on a \(k-\)abelian variety \(A\), is a morphism \(\lambda:A\longrightarrow A^{t}\) of degree \(d^{2}\), which equals the morphism \(\lambda_{L}:A\longrightarrow A^{t}\) for some ample line bundle \(L\) on \(A\). One can define in a similar way the notion of polarization of degree \(d\) on an abelian scheme \(A\) over an arbitrary scheme \(S\). We will denote \(\mathcal{A}_{g,d}\) the category fibered in groupoids over the category \(Sch\) of schemes whose fiber over \(S\) is the groupoid of abelian schemes over \(S\), of relative dimension \(g\), together with a polarization of degree \(d\). This is a Deligne-Mumford stack over \(\mathbb{Z}\), which is smooth over \(\mathbb{Z}[1/d]\), and which has a quasi-projective coarse moduli space.NEWLINENEWLINEOther intepretations of \(\mathcal{A}_{g,d}\) are introduced in section 2. A first one is due to \textit{V. Alexeev} [Ann. Math. (2) 155, No. 3, 611--708 (2002; Zbl 1052.14017)], and uses the notion of \(A-\)torsor for an abelian \(S-\)scheme \(A\): this is smooth \(S-\)scheme \(f:P\longrightarrow S\) having an action of \(A\) which is compatible with \(f\). Any relatively ample line bundle \(L\) on \(P\) induces a polarization \(\lambda_{L}\) on \(A\) whose degree is the rank of \(f_{*}L\). The Deligne-Mumford stack \(\mathcal{A}_{g}=\mathcal{A}_{g,1}\) is isomorphic to the category fibered in groupoids over \(Sch\) whose fiber over \(S\) is given by quadruples \((A,P,L,\theta)\) where \(A\) is an abelian \(S-\)scheme, \(P\) is an \(A-\)torsor, \(L\) is a relatively ample line bundle on \(P\) and \(\theta\) is a section of \(L\) over \(P\).NEWLINENEWLINEThe last interpretation uses theta groups and rigidifications of algebraic stacks. Namely let \(\mathcal{T}_{g,d}\) be the category fibered in groupoid over \(Sch\) whose fiber over \(S\) is given by triplets \((A,P,L)\) where \(A\) is an abelian \(S-\)scheme, \(P\) is an \(A-\)torsor and \(L\) is a relatively ample line bundle on \(P\) inducing a polarization of degree \(d\) on \(A\). Then \(\mathcal{A}_{g,d}\) is the rigidification of \(\mathcal{T}_{g,d}\) with respect to the closed substack \(\mathcal{G}\) of the inertia stack of \(\mathcal{T}_{g,d}\) whose fiber over \(S\) is given by the theta groups \(\mathcal{G}_{(A,P,L)}\).NEWLINENEWLINEThe different approaches to compactifications of \(\mathcal{A}_{g,d}\) are presented in section 4, and are based on degenerations of abelian varieties. To this purpose, the author reviews some basic results on semiabelian schemes, Fourier expansions and quadratic forms. One can then define toroidal compactifications \(\mathcal{A}_{g,\Sigma}\) over \(\mathrm{Spec}(\mathbb{Z})\), where \(\Sigma\) is an admissible (smooth) cone decomposition of the cone \(C(X)\) of positive semidefinite bilinear \(\mathbb{R}-\)forms on \(X\) whose radical is defined over \(\mathbb{Q}\). The toroidal compactification \(\mathcal{A}_{g,\Sigma}\) is an irreducible normal algebraic stack together with a dense open immersion \(j:\mathcal{A}_{g}\longrightarrow\mathcal{A}_{g,\Sigma}\) verifying some further properties related to \(\Sigma\) and to degenerations of abelian varieties. The choice of a particular \(\Sigma\), called second Voronoi decomposition, induces what is called the second Voronoi compactification \(\mathcal{A}_{g}^{\mathrm{Vor}}\) of \(\mathcal{A}_{g}\).NEWLINENEWLINEAnother compactification is due to Alexeev, and it is based on the description he gives of \(\mathcal{A}_{g}\), and which is based on degenerations of quadruples \((A,P,L,\theta)\): namely, one defines a category fibered in groupoids \(\mathcal{A}_{g}^{\mathrm{Alex}}\) over \(Sch\) whose fiber over \(S\) is given by quadruples \((G,P,L,\theta)\) where \(G\) is a semiabelian scheme, \(P\) is an \(S-\)scheme which is projective and flat with an \(S-\)action of \(G\), \(L\) is a relatively ample line bundle on \(S\) and \(\theta\) is a global section of \(L\) on \(P\), and such that the restriction of this quadruple to a geometric point of \(s\) is a stable semiabelic pair. The stack \(\mathcal{A}_{g}^{\mathrm{Alex}}\) is an Artin stack (in general non-irreducible) of finite type over \(\mathbb{Z}\), it has finite diagonal, and there is an open immersion \(j:\mathcal{A}_{g}\longrightarrow\mathcal{A}_{g}^{\mathrm{Alex}}\).NEWLINENEWLINEThe closure of the image of \(\mathcal{A}_{g}\) under this morphism is denoted \(\overline{\mathcal{A}}_{g}\), and [Compactifying moduli spaces for abelian varieties. Lecture Notes in Mathematics 1958. Berlin: Springer. (2008; Zbl 1165.14004)] gave a modular interpretation of it using logarithmic structures. It is indeed shown to be the rigidification of an algebraic stack \(\overline{\mathcal{T}}_{g}\) with respect to the subgroup \(\mathbb{G}_{m}\) of the inertia stack. Using a construction similar to that of \(\overline{\mathcal{T}}_{g}\), one defines an algebraic stack \(\overline{\mathcal{T}}_{g,d}\), whose rigidification \(\overline{\mathcal{A}}_{g,d}\) with respect to some closed substack of the inertia stack is a proper algebraic stack carrying a logarithmic structure, and it has a dense open immersion \(\mathcal{A}_{g,d}\longrightarrow\overline{\mathcal{A}}_{g,d}\).NEWLINENEWLINEFor the entire collection see [Zbl 1242.14002].