The cohomology groups of stable quasi-abelian schemes and degenerations associated with the $E_8$-lattice

Abstract

We study certain degenerate abelian schemes $(Q_0, L_0)$ that are GIT-stable in the sense that their SL-orbits are closed in the semistable locus. We prove the vanishing of the cohomology groups $H^q (Q_0, L_{0}^{n}) = 0$ for $q,n \gt 0$ for a naturally defined ample invertible sheaf $L_0$ on $Q_0$. When $n = 1$, this implies that $H^0 (Q_0, L_0)$, the space of global sections, is an irreducible module of the noncommutative Heisenberg group of $(Q_0, L_0)$.