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

We prove that the coherent cohomological dimension of the Siegel modular variety $A_{g,\mathrm{\Gamma }}$ is at most $g(g+1)\mathrm{/}2-2$ for $g\ge 2$. 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 $g\ge 2$ are automatically classical Siegel modular forms. Our result generalizes the work of Bruinier and Raum on the modularity of formal Siegel modular forms.