The paper under review studies the smoothness and singularities of two important toroidal compactifications of \(\mathcal A_g\), the moduli of principally polarized abelian varieties of dimension \(g\). A toroidal compactification of \(\mathcal A_g\) is a compactification constructed from an admissible rational polyhedral decomposition of \(\mathcal C_g\), the cone of positive-definite real quadratic forms over \(\mathbb R^g\). More precisely, the support of the decomposition is the rational closure \(\overline{\mathcal C}_g^{\mathrm{rc}}\) consisting of semi-positive-definite real quadratic forms with null spaces defined over \(\mathbb Q\). The two concerned compactifications are \(\mathcal A_g^{\mathrm{Perf}}\), the compactification associated to the perferct cone decomposition (also called the first Voronoi decomposition), and \(\mathcal A_g^{\mathrm{Vor}}\), the compactification associated to the second Voronoi decomposition. The main results are:
(Corollary 1.2) \(\mathcal A_g^{\mathrm{Perf}}\) is smooth for \(g\leqslant 3\) and the codimension of the singular locus and the non-simplicial locus are both \(10\) for \(g\geqslant 4\).
(Theorem 1.4) \(\mathcal A_g^{\mathrm{Vor}}\) is smooth for \(g\leqslant 4\) and the codimension of the non-simplicial locus is \(3\) for \(g\geqslant 5\).
Here what we mean by singularities are those “essential” singularities that can not be resolved simply by passing to the level covers. In other words, the singularities are the same with the toric singularities of the affine toric varieties for each rational polyhedral cone \(\sigma\subset \overline{\mathcal C}_g^{\mathrm{rc}}\) in the decompositions, and are independent on the actions of the arithmetic groups on \(\overline{\mathcal C}_g^{\mathrm{rc}}\). If we identify a quadratic form over \(\mathbb R^g\) with its associated symmetric matrix, and regard \(\mathcal C_g\) as a cone in the vector space of symmetric matrices \(\mathrm{Sym}^2(\mathbb R^g)\), then being smooth is equivalent to the corresponding monoid \(\sigma\cap \mathrm{Sym}^2(\mathbb Z^g)\) being generated by a subset of some \(\mathbb Z\)-basis of \(\mathrm{Sym}^2(\mathbb Z^g)\) (Such cones will be called basic.), and being simplicial (meaning the toric variety has abelian finite quotient singularities) is equivalent to the primitive vectors of the rays of \(\sigma\) being linearly independent over \(\mathbb R\) (Such cones will be called simplicial.). The main results thus follow from the following statements about the cone decompositions,
(Theorem 1.1) Every cone of dimension at most \(9\) in the perfect cone decomposition is basic. Moreover, with the exception of the cone of the root lattice lattice \(D_4\), every cone in the perfect cone decomposition of dimension at most \(10\) is simplicial.
(Theorem 1.4) For \(g\leqslant 4\), every cone in the second Voronoi decomposition is basic. For \(g\geqslant 5\), there are non-simplicial cones of dimension \(3\) in the second Voronoi decomposition.
Most of the paper is on the proof of Theorem 1.1. The proof can be understood without any background in the toroidal compactifications or the moduli of abelian varieties. The basic idea is to classify the cones of low dimensions (\(\leqslant 9\)) and check whether they are basic or simplicial by computers. However, when the dimension of the cone is \(10\), and the quadratic forms are over \(\mathbb R^g\) for \(g\geqslant 8\), a complete classification is not readily available. Then it is proved directly that the cones in those cases are always simplicial (Lemma 3.1). The proof is thus a combination of mathematical reasoning and computer-aided computations. Furthermore, a complete list of all \(9\)-dimensional perfect cones, which should be interesting on its own, is obtained during the proof and is provided in a separate electronic form.