Modularity of formal Fourier--Jacobi series from a cohomological point of view

We investigate the modularity of formal Fourier–Jacobi series by establishing cohomological vanishing results for line bundles defined on compactifications of ${\mathcal{A}}_g$. Working over $\mathbb{C}$, we show that the minimal compactification of ${\mathcal{A}}_2$ has only rational singularities, which allows us to characterize, for sufficiently large weights, the modularity of formal Fourier–Jacobi series of genus~$2$ via cohomological vanishing. Working over $\mathbb{Z}$, we introduce a level $n\ge 3$ structure and, via the the resolution morphism from a toroidal compactification of ${\mathcal{A}}_{g,n}$ to its minimal compactification, we characterize the modularity of arithmetic formal Fourier–Jacobi series with a level $n\ge 3$ structure and sufficiently large weight via cohomological vanishing.