Let $M_g$ be the coarse moduli space of smooth genus $g$ curves. Taking Jacobian induces an embedding $f\colon M_g\to A_g$, where $A_g$ is the coarse moduli space of $g$-dimensional prinicpally polarized abelian varieties. Let $\widetilde{A}_g$$F$ be the Borel-Baily compactification $A_g$. Leta locally free sheaf $\widetilde{M}_g$ be the closure of(or even coherent sheaf) on $M_g$ in $\widetilde{A}_g$, it is called the Satake compactification of $M_g$. (See GTM187, "Moduli of curves", p45)
Is $\widetilde{M}_g$ normal$H^i(M_g,F)$ always finite dimensional?
(We know that $\widetilde{M}_g$ coincides with $M_g$ in codim $1$. If $\widetilde{M}_g$ is normal (or equivalently,In the case $S_2$)$F=\mathcal{O}_{M_g}$, we can directly show $H^0(M_g,\mathcal{O}_{M_g})=\mathbb{C}$ by Hartogs theorem. I am curious if $H^i(M_g,F)$ is always finite dimensional for any coherent sheaf $F$ onknow $M_g$?$H^0(M_g,F)=\mathbb{C}$, see GTM187, "Moduli of curves", p45)
