Cuspidal modular forms - toroidal or minimal compactification?

Question

Let $Y$ be a Siegel variety and let $X$ be a toroidal compactification of $Y$.

For any tuple of integers $\underline k$ we have the usual sheaf $\omega^{\underline k}$. The space of modular forms of weight $k$ is $H^0(Y,\omega^{\underline k})= H^0(X,\omega^{\underline k})$ (let us suppose that the genus of $Y$ is greater than $1$). The space of cuspidal forms is defined as $H^0(X,\omega^{\underline k}(-D))$, where $D$ is the divisor of the boundary of $X$.

Let $X^\ast$ be the minimal compactification of $Y$ and let $\pi \colon X \to X^\ast$ be the canonical morphism. Of course we have $H^0(X,\omega^{\underline k}) = H^0(X^\ast,\pi_\ast\omega^{\underline k})$, so we can define modular forms as section of some sheaf on the minimal compactification (of course the sheaf need not to be locally free).

Let $I$ be the sheaf of ideals on $X^\ast$ that gives the boundary of $X^\ast$ (that is not a divisor in general). Is it true that we have $$ H^0(X,\omega^{\underline k}(-D)) = H^0(X^\ast,\pi_\ast\omega^{\underline k} \otimes I) ? $$ In other words, can we define cuspidal forms using the minimal compactification?

A related question is the following: can we consider the space $H^0(X,\omega^{\underline k} \otimes \pi^\ast I)$?

Note that $I$ is not locally free, so we cannot use the projection formula.

Thank you!

Answer 1

Let $i \colon \Delta \to X^\ast$ be the closed immersion corresponding to the boundary of $X^\ast$. You have an exact sequence $$ 0 \to I \to \mathcal O_{X^\ast} \to i_\ast \mathcal O_\Delta \to 0 $$ but, as you said, $\pi_\ast \omega^{\underline k}$ is not locally free in general, so, after tensoring with $\pi_\ast \omega^{\underline k}$ we only have the exact sequence $$ I \otimes \pi_\ast \omega^{\underline k} \to \pi_\ast \omega^{\underline k} \to i_\ast \mathcal O_\Delta \otimes \pi_\ast \omega^{\underline k} \to 0 $$ so the correct sheaf to consider is $$ \ker(\pi_\ast \omega^{\underline k} \to i_\ast \mathcal O_\Delta \otimes \pi_\ast \omega^{\underline k}). $$