[Paper Review] The singularities of $A_g^P$

Judith A. Armstrong, N. I. Shepherd‐Barron|arXiv (Cornell University)|Apr 20, 2016

Algebraic Geometry and Number Theory1 references3 citations

TL;DR

This paper completes the proof that the perfect cone compactification $A_g^P$ of the moduli space of principally polarized abelian $g$-folds has canonical singularities for $g = 5$ and terminal singularities for $g \geq 6$ in characteristic zero. It corrects a gap in earlier work by strengthening a key proposition using representation-theoretic analysis of group actions on tangent spaces via the Riemann–Shimizu–Todorov (RST) criterion.

ABSTRACT

We complete the proof given earlier by one of us that the singularities of the perfect cone compactification $A_g^P$ of the coarse moduli space $A_g$ of principally polarized abelian $g$-folds are terminal if $g\ge 6$, and canonical if $g=5$.

Motivation & Objective

• To complete the proof of the singularities type of $A_g^P$ for $g \geq 5$, correcting an incomplete argument in prior work.
• To establish that $A_g^P$ has canonical singularities when $g = 5$ and terminal singularities when $g \geq 6$ in characteristic zero.
• To resolve a gap in Shepherd-Barron's earlier proof by strengthening Proposition 3.2 using representation-theoretic analysis of group actions on tangent spaces.
• To apply the Riemann–Shimizu–Todorov (RST) criterion to determine singularity types via fractional parts of eigenvalues of group elements.

Proposed method

• Apply the RST criterion to analyze singularities of the quotient stack $[\mathcal{X}/G]$, where $\mathcal{X}$ is a $G$-equivariant bundle over a torus embedding.
• Define $\lambda_{Y,P}(s)$ as the sum of $\arg(\zeta)/2\pi$ over eigenvalues $\zeta$ of $s$ on the tangent space $T_{Y,P}$, which determines singularity type.
• Use the action of a finite group $G$ on a smooth variety $Y$ to reduce the problem to checking $\lambda_{Y,P}(s) \geq 1$ (canonical) or $>1$ (terminal) for all non-identity $s \in G$.
• Analyze the representation of $G$ on $\mathrm{Sym}^2 W \oplus (W \otimes \Lambda)$, where $W = H^0(C, \Omega^1_C)^\vee$ and $\Lambda = \mathbb{Z}^r$, to compute $\lambda$-values.
• Use a $G$-equivariant resolution $\widetilde{X} \to X$ and analyze the action on exceptional divisors to ensure $G$ acts freely in codimension one.
• Apply a theorem of Snurnikov to extend singularity type results from the open torus to the full compactification when $h = 0$.

Experimental results

Research questions

• RQ1 Does the perfect cone compactification $A_g^P$ have canonical singularities for $g = 5$?
• RQ2 Does $A_g^P$ have terminal singularities for $g \geq 6$ in characteristic zero?
• RQ3 Is the earlier proof of this result in [SB] complete, or are there gaps in the argument?
• RQ4 Can the RST criterion be applied effectively to toroidal compactifications of $A_g^P$ via group representation analysis?
• RQ5 How do the eigenvalues of group elements acting on tangent spaces determine the type of singularities?

Key findings

• The singularities of $A_g^P$ are canonical when $g = 5$ and terminal when $g \geq 6$ in characteristic zero.
• The proof is completed by replacing an insufficiently strong proposition in [SB] with a refined analysis of $\lambda$-values for group actions on $\mathrm{Sym}^2 W$ and $V = \mathrm{Sym}^2 W \oplus (W \otimes \Lambda)$.
• For $g \geq 6$, $\lambda_{\mathrm{Sym}^2 W}(s) > 1$ whenever $s|_W \neq \pm 1$, ensuring terminal singularities.
• When $\lambda_V(s) < 1$, the only possibility is $h = 1$, $s|_W = -1$, $s|_{\Lambda_\mathbb{Q}} = (1, (-1)^{r-1})$, and $\lambda_V(s) = 1/2$, which is ruled out in higher $g$.
• The case $h = 0$ is resolved using a theorem of Snurnikov, extended here with a self-contained proof for $G$-equivariant torus embeddings.
• The $G$-action is shown to be free in codimension one on the resolution, ensuring the RST criterion applies and singularity types are preserved.