Metrics and Coordinate Rings on Voronoi Compactifications

Definitions and mathematical statement

Let $\mathcal A_g$ be the moduli space or stack of principally polarized complex abelian varieties of dimension $g$, and let $\mathcal A_g^F$ denote its first Voronoi, or perfect-cone, compactification. The source introduces a universal family $U_g\to\mathcal A_g$, a compactification $\overline U_g$ identified with the boundary divisor in $\mathcal A_{g+1}^F$, and the restricted boundary line bundle. To distinguish the two ambient genera, write that boundary as $D_{g+1}$ and set \[ L=\mathcal O(D_{g+1})|_{\overline U_g}. \] These objects require a stack or level convention: on a coarse moduli space a universal abelian scheme need not exist, and a boundary may be a Kummer quotient rather than literally the universal family. The source's identification, or the pullback through its intended covering map, must be checked before it is used as an equality of schemes and line bundles.

The first question is whether $L$ admits a Hermitian metric $h$ whose curvature on the intended generic smooth abelian fiber $A$ is translation invariant: \[ t_a^*(\Theta_h|_A)=\Theta_h|_A\qquad(a\in A). \] Here $t_a$ is translation by $a$. The requested regularity of the metric across the compactification and whether singular metrics are allowed are not stated. No positivity of curvature is requested. A metric on each separate fiber is not by itself a metric extending on $L$ over $\overline U_g$; that global extension is part of the question.

For the second question, let $D_g$ be the boundary divisor on $\mathcal A_g^F$, and let $M$ be the divisor or line-bundle class used in the source, conventionally the modular-form or Hodge class but not defined in this record. The two candidate rings are the total coordinate ring $\operatorname{TC}(\mathcal A_g^F)$ and the displayed subring \[ R_g=\bigoplus_{a,b\geq0}H^0(\mathcal A_g^F,\mathcal O(aM-bD_g)), \] with multiplication adding the pair $(a,b)$. Are these rings finitely generated over ${\mathbb{C}}$? For $R_g$, this means finitely many homogeneous sections generate all bidegrees. The total coordinate ring needs a specified divisor-class grading and convention; if classes are only ${\mathbb{Q}}$-Cartier, the meaning of each sheaf or allowed multiples must also be fixed. The two finite-generation questions are not asserted equivalent, and no bound on the generators uniform in $g$ is requested.

The supplied assessment checked 2026-08-21 reports substantial partial results on the metric and finite-generation questions, without a complete resolution of their combined general formulation. No exact genus range or theorem hypotheses are reproduced.

Short English statement

Ask for a global metric on the boundary line bundle with translation-invariant curvature on the generic abelian fiber, and for finite generation of the indicated total coordinate ring or modular-form subring. Stack, level, divisor, and metric-extension conventions are essential.

Sources

• Local UnsolvedMath registry, permanent ID 30000188, catalogue entry, Metrics and Coordinate Rings on Voronoi Compactifications.
• Komplexe Algebraische Geometrie (2005), Oberwolfach Reports, pp. 407--463, DOI https://doi.org/10.4171/owr/2005/07.
• Shepherd-Barron, supplied label "perfect forms and the moduli space of abelian varieties": https://arxiv.org/abs/math/0502362.
• Supplied primary-source verification says that the family, compactification, boundary, and line bundle are introduced immediately before the two questions; supplied literature triage checked 2026-08-21.