On the cohomological dimension of Siegel modular varieties and the modularity of formal Siegel modular forms

TL;DR

The paper proves that the coherent cohomological dimension of the Siegel modular variety $A_{g,\Gamma}$ satisfies $\mathrm{ccd}(A_{g,\Gamma}) \le d-2$ with $d=\frac{1}{2}g(g+1)$ for $g\ge 2$, which implies the boundary satisfies the Grothendieck–Lefschetz condition and that formal Siegel modular forms are automatically classical. Central to the argument is the development of weak $G2$/$G3$ properties relative to a line bundle, their functorial behavior under morphisms and finite covers, and the connection to Lefschetz-type vanishing. By applying these tools to the minimal and toroidal compactifications of Siegel modular varieties, the authors establish the Lefschetz property and deduce the desired cohomological bound, which in turn yields convergence and modularity results for formal Siegel modular forms. The work generalizes Bruinier–Raum’s modularity results and provides a framework that could extend to broader Shimura varieties with suitable rank conditions, offering new avenues for automatic convergence and modularity phenomena in arithmetic geometry.

Abstract

We prove that the coherent cohomological dimension of the Siegel modular variety $A_{g,Γ}$ is at most $g(g+1)/2-2$ for $g\geq 2$. As a corollary, we show that the boundary of the compactified Siegel modular variety satisfies the Grothendieck-Lefschetz condition. This implies, in particular, that formal Siegel modular forms of genus $g\geq2$ are automatically classical Siegel modular forms. Our result generalizes the work of Bruinier and Raum on the modularity of formal Siegel modular forms.

Theorems & Definitions (71)

• Theorem 1.1: Theorem \ref{['ccd_bound']}
• Conjecture 1.2
• Definition 1.3
• Definition 1.4
• Theorem 1.5: Corollary \ref{['weak_g3_l']}
• Definition 1.6
• Theorem 1.7: Corollary \ref{['lef_l']}
• Definition 2.1
• Proposition 2.2
• Definition 2.3