Intersection Numbers on First Voronoi Compactifications

Definitions and mathematical statement

For a positive integer $g$, let $\mathcal A_g$ be the moduli space of principally polarized abelian varieties of dimension $g$, and let $\mathcal A_g^F$ be its first Voronoi, or perfect cone, compactification. Put $N=g(g+1)/2=\dim\mathcal A_g^F$. Let $L$ be the Hodge divisor class, conventionally the first Chern class of the determinant Hodge bundle, and let $D$ be the source's boundary divisor class. With the appropriate rational intersection-theory convention define \[a_n^{(g)}=\deg_{\mathcal A_g^F}\left(L^nD^{N-n}\right),\qquad n\in{\mathbb{Z}},\quad0\leq n\leq N.\] The conjecture is \[n\notin\left\{\frac{k(k+1)}2:k\in{\mathbb{Z}},\ 0\leq k\leq g\right\}\quad\Longrightarrow\quad a_n^{(g)}=0.\] Equivalently, a nonzero top intersection number may occur only at a triangular exponent of $L$. This is only a necessary condition for nonvanishing; it does not assert that every listed triangular exponent gives a nonzero value. The compactification is the first Voronoi/perfect cone one, not a different toroidal compactification. Stack versus coarse-space normalization and the boundary multiplicity convention are not specified; their role should be confirmed even though the target here is vanishing. The supplied 2026-08-21 triage labels the assertion open and reports computations in substantial ranges and low genus without a general proof.

Short English statement

Must every top Hodge-boundary intersection vanish unless the exponent of the Hodge class is a triangular number? The question gives a necessary pattern for nonvanishing, without asserting its converse.

Sources

• Local UnsolvedMath registry, permanent ID 30001017, source problem catalogue entry, Intersection Numbers on First Voronoi Compactifications, proposed year 2008; no individual proposer supplied.
• Supplied citation: Komplexe Analysis (2009), Oberwolfach Reports, DOI 10.4171/owr/2008/38, pp. 2165--2218: https://doi.org/10.4171/owr/2008/38. Supplied literature label: original report.
• Supplied literature label: Erdenberger, Grushevsky and Hulek, https://arxiv.org/abs/0707.1274.
• Literature assessment date: 2026-08-21. The status and progress claims are supplied dated curation, not independently verified here.